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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm2mpfval 20601* A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( T ` M ) = ( Q gsumg ( k e. NN0 |-> ( ( M decompPMat k ) .* ( k .^ X ) ) ) ) )
 
Theorempm2mpcl 20602 The transformation of polynomial matrices into polynomials over matrices maps polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   &    |- L = ( Base ` Q )   =>    |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. L )
 
Theorempm2mpf 20603 The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   &    |- L = ( Base ` Q )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T : B --> L )
 
Theorempm2mpf1 20604 The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   &    |- L = ( Base ` Q )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T : B -1-1-> L )
 
Theorempm2mpcoe1 20605 A coefficient of the polynomial over matrices which is the result of the transformation of a polynomial matrix is the matrix consisting of the coefficients in the polynomial entries of the polynomial matrix. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( M e. B /\ K e. NN0 ) ) -> ( (coe1 ` ( T ` M ) ) ` K ) = ( M decompPMat K ) )
 
Theoremidpm2idmp 20606 The transformation of the identity polynomial matrix into polynomials over matrices results in the identity of the polynomials over matrices. (Contributed by AV, 18-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> ( T ` ( 1r ` C ) ) = ( 1r ` Q ) )
 
Theoremmptcoe1matfsupp 20607* The mapping extracting the entries of the coefficient matrices of a polynomial over matrices at a fixed position is finitely supported. (Contributed by AV, 6-Oct-2019.) (Proof shortened by AV, 23-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ I e. N /\ J e. N ) -> ( k e. NN0 |-> ( I ( (coe1 ` O ) ` k ) J ) ) finSupp ( 0g ` R ) )
 
Theoremmply1topmatcllem 20608* Lemma for mply1topmatcl 20610. (Contributed by AV, 6-Oct-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- P = (Poly1 ` R )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ I e. N /\ J e. N ) -> ( k e. NN0 |-> ( ( I ( (coe1 ` O ) ` k ) J ) .x. ( k E Y ) ) ) finSupp ( 0g ` P ) )
 
Theoremmply1topmatval 20609* A polynomial over matrices transformed into a polynomial matrix. I is the inverse function of the transformation T of polynomial matrices into polynomials over matrices: ( T ` ( I ` O ) ) = O ) (see mp2pm2mp 20616). (Contributed by AV, 6-Oct-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- P = (Poly1 ` R )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   =>    |- ( ( N e. V /\ O e. L ) -> ( I ` O ) = ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )
 
Theoremmply1topmatcl 20610* A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- P = (Poly1 ` R )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   =>    |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( I ` O ) e. B )
 
Theoremmp2pm2mplem1 20611* Lemma 1 for mp2pm2mp 20616. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   =>    |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( I ` O ) = ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )
 
Theoremmp2pm2mplem2 20612* Lemma 2 for mp2pm2mp 20616. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   =>    |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) ) ) ) e. B )
 
Theoremmp2pm2mplem3 20613* Lemma 3 for mp2pm2mp 20616. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- P = (Poly1 ` R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ K e. NN0 ) -> ( ( I ` O ) decompPMat K ) = ( i e. N , j e. N |-> ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) ) ) ) ` K ) ) )
 
Theoremmp2pm2mplem4 20614* Lemma 4 for mp2pm2mp 20616. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- P = (Poly1 ` R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ K e. NN0 ) -> ( ( I ` O ) decompPMat K ) = ( (coe1 ` O ) ` K ) )
 
Theoremmp2pm2mplem5 20615* Lemma 5 for mp2pm2mp 20616. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- P = (Poly1 ` R )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   =>    |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( k e. NN0 |-> ( ( ( I ` O ) decompPMat k ) .* ( k .^ X ) ) ) finSupp ( 0g ` Q ) )
 
Theoremmp2pm2mp 20616* A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)
|- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )   &    |- P = (Poly1 ` R )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( T ` ( I ` O ) ) = O )
 
Theorempm2mpghmlem2 20617* Lemma 2 for pm2mpghm 20621. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   =>    |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( k e. NN0 |-> ( ( M decompPMat k ) .* ( k .^ X ) ) ) finSupp ( 0g ` Q ) )
 
Theorempm2mpghmlem1 20618 Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ K e. NN0 ) -> ( ( M decompPMat K ) .* ( K .^ X ) ) e. L )
 
Theorempm2mpfo 20619 The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T : B -onto-> L )
 
Theorempm2mpf1o 20620 The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 14-Oct-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T : B -1-1-onto-> L )
 
Theorempm2mpghm 20621 The transformation of polynomial matrices into polynomials over matrices is an additive group homomorphism. (Contributed by AV, 16-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( C GrpHom Q ) )
 
Theorempm2mpgrpiso 20622 The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( C GrpIso Q ) )
 
Theorempm2mpmhmlem1 20623* Lemma 1 for pm2mpmhm 20625. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- L = ( Base ` Q )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( l e. NN0 |-> ( ( A gsumg ( k e. ( 0 ... l ) |-> ( ( x decompPMat k ) ( .r ` A ) ( y decompPMat ( l - k ) ) ) ) ) .* ( l .^ X ) ) ) finSupp ( 0g ` Q ) )
 
Theorempm2mpmhmlem2 20624* Lemma 2 for pm2mpmhm 20625. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   &    |- B = ( Base ` C )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> A. x e. B A. y e. B ( T ` ( x ( .r ` C ) y ) ) = ( ( T ` x ) ( .r ` Q ) ( T ` y ) ) )
 
Theorempm2mpmhm 20625 The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( (mulGrp ` C ) MndHom (mulGrp ` Q ) ) )
 
Theorempm2mprhm 20626 The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( C RingHom Q ) )
 
Theorempm2mprngiso 20627 The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   &    |- T = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( C RingIso Q ) )
 
Theorempmmpric 20628 The ring of polynomial matrices over a ring is isomorphic to the ring of polynomials over matrices of the same dimension over the same ring. (Contributed by AV, 30-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- A = ( N Mat R )   &    |- Q = (Poly1 ` A )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> C ~=r Q )
 
Theoremmonmat2matmon 20629 The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-Nov-2019.) (Revised by AV, 7-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- K = ( Base ` A )   &    |- Q = (Poly1 ` A )   &    |- I = ( N pMatToMatPoly R )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- .x. = ( .s ` C )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. K /\ L e. NN0 ) ) -> ( I ` ( ( L E Y ) .x. ( T ` M ) ) ) = ( M .* ( L .^ X ) ) )
 
Theorempm2mp 20630* The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 7-Dec-2019.)
|- P = (Poly1 ` R )   &    |- C = ( N Mat P )   &    |- B = ( Base ` C )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- X = (var1 ` A )   &    |- A = ( N Mat R )   &    |- K = ( Base ` A )   &    |- Q = (Poly1 ` A )   &    |- I = ( N pMatToMatPoly R )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Y = (var1 ` R )   &    |- .x. = ( .s ` C )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. ( K ^m NN0 ) /\ M finSupp ( 0g ` A ) ) ) -> ( I ` ( C gsumg ( n e. NN0 |-> ( ( n E Y ) .x. ( T ` ( M ` n ) ) ) ) ) ) = ( Q gsumg ( n e. NN0 |-> ( ( M ` n ) .* ( n .^ X ) ) ) ) )
 
11.5  The characteristic polynomial

According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients.". Based on the definition of the characteristic polynomial of a square matrix (df-chpmat 20632) the eigenvalues and corresponding eigenvectors can be defined.

 
11.5.1  Definition and basic properties

The characteristic polynomial of a matrix A is the determinat of the characteristic matrix of A: ( t I - A ).

 
Syntaxcchpmat 20631 Extend class notation with the characteristic polynomial.
class CharPlyMat
 
Definitiondf-chpmat 20632* Define the characteristic polynomial of a square matrix. According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "The characteristic polynomial of [an n x n matrix] A, denoted by pA(t), is the polynomial defined by pA ( t ) = det ( t I - A ) where I denotes the n-by-n identity matrix.". In addition, however, the underlying ring must be commutative, see definition in [Lang], p. 561: " Let k be a commutative ring ... Let M be any n x n matrix in k ... We define the characteristic polynomial PM(t) to be the determinant det ( t In - M ) where In is the unit n x n matrix." To be more precise, the matrices A and I on the right hand side are matrices with coefficients of a polynomial ring. Therefore, the original matrix A over a given commutative ring must be transformed into corresponding matrices over the polynomial ring over the given ring. (Contributed by AV, 2-Aug-2019.)
|- CharPlyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) )
 
Theoremchmatcl 20633 Closure of the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) (Proof shortened by AV, 29-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- T = ( N matToPolyMat R )   &    |- .- = ( -g ` Y )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- H = ( ( X .x. .1. ) .- ( T ` M ) )   =>    |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> H e. ( Base ` Y ) )
 
Theoremchmatval 20634 The entries of the characteristic matrix of a matrix. (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 10-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- T = ( N matToPolyMat R )   &    |- .- = ( -g ` Y )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- H = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- .~ = ( -g ` P )   &    |- .0. = ( 0g ` P )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I H J ) = if ( I = J , ( X .~ ( I ( T ` M ) J ) ) , ( .0. .~ ( I ( T ` M ) J ) ) ) )
 
Theoremchpmatfval 20635* Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- D = ( N maDet P )   &    |- .- = ( -g ` Y )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- T = ( N matToPolyMat R )   &    |- .1. = ( 1r ` Y )   =>    |- ( ( N e. Fin /\ R e. V ) -> C = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )
 
Theoremchpmatval 20636 The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- D = ( N maDet P )   &    |- .- = ( -g ` Y )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- T = ( N matToPolyMat R )   &    |- .1. = ( 1r ` Y )   =>    |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( C ` M ) = ( D ` ( ( X .x. .1. ) .- ( T ` M ) ) ) )
 
Theoremchpmatply1 20637 The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.)
|- C = ( N CharPlyMat R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- E = ( Base ` P )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( C ` M ) e. E )
 
Theoremchpmatval2 20638* The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .- = ( -g ` Y )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- T = ( N matToPolyMat R )   &    |- .1. = ( 1r ` Y )   &    |- G = ( SymGrp ` N )   &    |- H = ( Base ` G )   &    |- Z = ( ZRHom ` P )   &    |- S = (pmSgn ` N )   &    |- U = (mulGrp ` P )   &    |- .X. = ( .r ` P )   =>    |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( C ` M ) = ( P gsumg ( p e. H |-> ( ( ( Z o. S ) ` p ) .X. ( U gsumg ( x e. N |-> ( ( p ` x ) ( ( X .x. .1. ) .- ( T ` M ) ) x ) ) ) ) ) ) )
 
Theoremchpmat0d 20639 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
|- C = ( (/) CharPlyMat R )   =>    |- ( R e. Ring -> ( C ` (/) ) = ( 1r ` (Poly1 ` R ) ) )
 
Theoremchpmat1dlem 20640 Lemma for chpmat1d 20641. (Contributed by AV, 7-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .- = ( -g ` P )   &    |- S = (algSc ` P )   &    |- G = ( N Mat P )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( R e. Ring /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I ( ( X ( .s ` G ) ( 1r ` G ) ) ( -g ` G ) ( T ` M ) ) I ) = ( X .- ( S ` ( I M I ) ) ) )
 
Theoremchpmat1d 20641 The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .- = ( -g ` P )   &    |- S = (algSc ` P )   =>    |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( C ` M ) = ( X .- ( S ` ( I M I ) ) ) )
 
Theoremchpdmatlem0 20642 Lemma 0 for chpdmat 20646. (Contributed by AV, 18-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- S = (algSc ` P )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .0. = ( 0g ` R )   &    |- G = (mulGrp ` P )   &    |- .- = ( -g ` P )   &    |- Q = ( N Mat P )   &    |- .1. = ( 1r ` Q )   &    |- .x. = ( .s ` Q )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> ( X .x. .1. ) e. ( Base ` Q ) )
 
Theoremchpdmatlem1 20643 Lemma 1 for chpdmat 20646. (Contributed by AV, 18-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- S = (algSc ` P )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .0. = ( 0g ` R )   &    |- G = (mulGrp ` P )   &    |- .- = ( -g ` P )   &    |- Q = ( N Mat P )   &    |- .1. = ( 1r ` Q )   &    |- .x. = ( .s ` Q )   &    |- Z = ( -g ` Q )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( ( X .x. .1. ) Z ( T ` M ) ) e. ( Base ` Q ) )
 
Theoremchpdmatlem2 20644 Lemma 2 for chpdmat 20646. (Contributed by AV, 18-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- S = (algSc ` P )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .0. = ( 0g ` R )   &    |- G = (mulGrp ` P )   &    |- .- = ( -g ` P )   &    |- Q = ( N Mat P )   &    |- .1. = ( 1r ` Q )   &    |- .x. = ( .s ` Q )   &    |- Z = ( -g ` Q )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ i e. N ) /\ j e. N ) /\ i =/= j ) /\ ( i M j ) = .0. ) -> ( i ( ( X .x. .1. ) Z ( T ` M ) ) j ) = ( 0g ` P ) )
 
Theoremchpdmatlem3 20645 Lemma 3 for chpdmat 20646. (Contributed by AV, 18-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- S = (algSc ` P )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .0. = ( 0g ` R )   &    |- G = (mulGrp ` P )   &    |- .- = ( -g ` P )   &    |- Q = ( N Mat P )   &    |- .1. = ( 1r ` Q )   &    |- .x. = ( .s ` Q )   &    |- Z = ( -g ` Q )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ K e. N ) -> ( K ( ( X .x. .1. ) Z ( T ` M ) ) K ) = ( X .- ( S ` ( K M K ) ) ) )
 
Theoremchpdmat 20646* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- S = (algSc ` P )   &    |- B = ( Base ` A )   &    |- X = (var1 ` R )   &    |- .0. = ( 0g ` R )   &    |- G = (mulGrp ` P )   &    |- .- = ( -g ` P )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) -> ( C ` M ) = ( G gsumg ( k e. N |-> ( X .- ( S ` ( k M k ) ) ) ) ) )
 
Theoremchpscmat 20647* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }   &    |- S = (algSc ` P )   &    |- .- = ( -g ` P )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ I e. N /\ A. n e. N ( n M n ) = E ) ) -> ( C ` M ) = ( ( # ` N ) .^ ( X .- ( S ` E ) ) ) )
 
Theoremchpscmat0 20648* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }   &    |- S = (algSc ` P )   &    |- .- = ( -g ` P )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ I e. N /\ A. n e. N ( n M n ) = ( I M I ) ) ) -> ( C ` M ) = ( ( # ` N ) .^ ( X .- ( S ` ( I M I ) ) ) ) )
 
Theoremchpscmatgsumbin 20649* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }   &    |- S = (algSc ` P )   &    |- .- = ( -g ` P )   &    |- F = (.g ` P )   &    |- H = (mulGrp ` R )   &    |- E = (.g ` H )   &    |- I = ( invg ` R )   &    |- .x. = ( .s ` P )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( C ` M ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) ) ) )
 
Theoremchpscmatgsummon 20650* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }   &    |- S = (algSc ` P )   &    |- .- = ( -g ` P )   &    |- F = (.g ` P )   &    |- H = (mulGrp ` R )   &    |- E = (.g ` H )   &    |- I = ( invg ` R )   &    |- .x. = ( .s ` P )   &    |- Z = (.g ` R )   =>    |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( C ` M ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) ) ) )
 
Theoremchp0mat 20651 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- .0. = ( 0g ` A )   =>    |- ( ( N e. Fin /\ R e. CRing ) -> ( C ` .0. ) = ( ( # ` N ) .^ X ) )
 
Theoremchpidmat 20652 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
|- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- A = ( N Mat R )   &    |- X = (var1 ` R )   &    |- G = (mulGrp ` P )   &    |- .^ = (.g ` G )   &    |- I = ( 1r ` A )   &    |- S = (algSc ` P )   &    |- .1. = ( 1r ` R )   &    |- .- = ( -g ` P )   =>    |- ( ( N e. Fin /\ R e. CRing ) -> ( C ` I ) = ( ( # ` N ) .^ ( X .- ( S ` .1. ) ) ) )
 
Theoremchmaidscmat 20653 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 19-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- E = ( Base ` P )   &    |- Y = ( N Mat P )   &    |- K = ( Base ` Y )   &    |- .x. = ( .s ` Y )   &    |- .0. = ( 0g ` P )   &    |- .1. = ( 1r ` Y )   &    |- S = ( N ScMat P )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( C ` M ) .x. .1. ) e. S )
 
11.5.2  The characteristic factor function G

In this subsection the function G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) ) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 20683. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 20671, cayhamlem3 20692 and cayhamlem4 20693.

 
Theoremfvmptnn04if 20654* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )   &    |- ( ph -> S e. NN )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> Y e. V )   &    |- ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )   &    |- ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )   &    |- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )   &    |- ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )   =>    |- ( ph -> ( G ` N ) = Y )
 
Theoremfvmptnn04ifa 20655* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )   &    |- ( ph -> S e. NN )   &    |- ( ph -> N e. NN0 )   =>    |- ( ( ph /\ N = 0 /\ [_ N / n ]_ A e. V ) -> ( G ` N ) = [_ N / n ]_ A )
 
Theoremfvmptnn04ifb 20656* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )   &    |- ( ph -> S e. NN )   &    |- ( ph -> N e. NN0 )   =>    |- ( ( ph /\ ( 0 < N /\ N < S ) /\ [_ N / n ]_ B e. V ) -> ( G ` N ) = [_ N / n ]_ B )
 
Theoremfvmptnn04ifc 20657* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )   &    |- ( ph -> S e. NN )   &    |- ( ph -> N e. NN0 )   =>    |- ( ( ph /\ N = S /\ [_ N / n ]_ C e. V ) -> ( G ` N ) = [_ N / n ]_ C )
 
Theoremfvmptnn04ifd 20658* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )   &    |- ( ph -> S e. NN )   &    |- ( ph -> N e. NN0 )   =>    |- ( ( ph /\ S < N /\ [_ N / n ]_ D e. V ) -> ( G ` N ) = [_ N / n ]_ D )
 
Theoremchfacfisf 20659* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> G : NN0 --> ( Base ` Y ) )
 
Theoremchfacfisfcpmat 20660* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- S = ( N ConstPolyMat R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> G : NN0 --> S )
 
Theoremchfacffsupp 20661* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> G finSupp ( 0g ` Y ) )
 
Theoremchfacfscmulcl 20662* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- .^ = (.g ` (mulGrp ` P ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) /\ K e. NN0 ) -> ( ( K .^ X ) .x. ( G ` K ) ) e. ( Base ` Y ) )
 
Theoremchfacfscmul0 20663* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- .^ = (.g ` (mulGrp ` P ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) /\ K e. ( ZZ>= ` ( s + 2 ) ) ) -> ( ( K .^ X ) .x. ( G ` K ) ) = .0. )
 
Theoremchfacfscmulfsupp 20664* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- .^ = (.g ` (mulGrp ` P ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( i e. NN0 |-> ( ( i .^ X ) .x. ( G ` i ) ) ) finSupp .0. )
 
Theoremchfacfscmulgsum 20665* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- X = (var1 ` R )   &    |- .x. = ( .s ` Y )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .+ = ( +g ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( Y gsumg ( i e. NN0 |-> ( ( i .^ X ) .x. ( G ` i ) ) ) ) = ( ( Y gsumg ( i e. ( 1 ... s ) |-> ( ( i .^ X ) .x. ( ( T ` ( b ` ( i - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` i ) ) ) ) ) ) ) .+ ( ( ( ( s + 1 ) .^ X ) .x. ( T ` ( b ` s ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) ) )
 
Theoremchfacfpmmulcl 20666* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) /\ K e. NN0 ) -> ( ( K .^ ( T ` M ) ) .X. ( G ` K ) ) e. ( Base ` Y ) )
 
Theoremchfacfpmmul0 20667* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) /\ K e. ( ZZ>= ` ( s + 2 ) ) ) -> ( ( K .^ ( T ` M ) ) .X. ( G ` K ) ) = .0. )
 
Theoremchfacfpmmulfsupp 20668* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( i e. NN0 |-> ( ( i .^ ( T ` M ) ) .X. ( G ` i ) ) ) finSupp .0. )
 
Theoremchfacfpmmulgsum 20669* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   &    |- .+ = ( +g ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( Y gsumg ( i e. NN0 |-> ( ( i .^ ( T ` M ) ) .X. ( G ` i ) ) ) ) = ( ( Y gsumg ( i e. ( 1 ... s ) |-> ( ( i .^ ( T ` M ) ) .X. ( ( T ` ( b ` ( i - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` i ) ) ) ) ) ) ) .+ ( ( ( ( s + 1 ) .^ ( T ` M ) ) .X. ( T ` ( b ` s ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) ) )
 
Theoremchfacfpmmulgsum2 20670* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   &    |- .+ = ( +g ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( Y gsumg ( i e. NN0 |-> ( ( i .^ ( T ` M ) ) .X. ( G ` i ) ) ) ) = ( ( Y gsumg ( i e. ( 1 ... s ) |-> ( ( ( i .^ ( T ` M ) ) .X. ( T ` ( b ` ( i - 1 ) ) ) ) .- ( ( ( i + 1 ) .^ ( T ` M ) ) .X. ( T ` ( b ` i ) ) ) ) ) ) .+ ( ( ( ( s + 1 ) .^ ( T ` M ) ) .X. ( T ` ( b ` s ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) ) )
 
Theoremcayhamlem1 20671* Lemma 1 for cayleyhamilton 20695. (Contributed by AV, 11-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- .^ = (.g ` (mulGrp ` Y ) )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( Y gsumg ( i e. NN0 |-> ( ( i .^ ( T ` M ) ) .X. ( G ` i ) ) ) ) = .0. )
 
11.5.3  The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * In - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * In - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * In - A ) x B = det(t * In - A) x In = p(t) * In .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix Bi of numbers, such that one has

(3) B = sumi = 0 to (n-1) t^i Bi .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * In = ( t * In - A ) x B
= ( t * In - A ) x sumi = 0 to (n-1) t^i * Bi
 = sumi = 0 to (n-1) t * In x t^i Bi - sumi = 0 to (n-1) A * t^i Bi
 = sumi = 0 to (n-1) t^(i+1) * Bi - sumi = 0 to (n-1) t^i * A x Bi
 = t^n Bn-1 + sumi = 1 to (n-1) t^i * ( Bi-1 - A x Bi ) - A x B0 .

Writing

(5) p(t) In = t^n * In + t^(n-1) * c(n-1) x In + ... + t * c1 In + c0 In ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) Bn-1 = In , Bi-1 - A x Bi = ci * In for 1 <= i <= n-1 , - A x B0 = c0 * In .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n Bn-1 + sumi = 1 to (n-1) ( A^i x Bi-1 - A^(i+1) x Bi ) - A x B0 = A^n + cn-1 * A^(n-1) + ... + c1 * A + c0 * In .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and In is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * In - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * In - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 20632, the characteristic polynomial of an N x N matrix M over a ring R is defined as

( ( N CharPlyMat R ) ` M ) = ( D ` ( ( X .x. .1. ) .- ( T ` M ) ) ) )

where D = ( N maDet P ) is the function mapping an N x N matrix over the polynomial ring over the ring R to its determinant, X = (var1 ` R ) is the variable of the polynomials over R, .1. is the N x N identity matrix as matrix over the polynomial ring over the ring R (not the N x N identity matrix of the matrices over the ring R!) and ( T ` M ) = ( ( N matToPolyMat R ) ` M ) is the matrix M over a ring R transformed into a constant matrix over the polynomial ring over the ring R. Thus .x. is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 20636).

By this definition, it is ensured that ( ( X .x. .1. ) .- ( T ` M ) ), the matrix whose determinat is the characteristic polynomial of the matrix M, is actually a matrix over the polynomial ring over the ring R, as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix M (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix M, formally defined by ( ( N CharPlyMat R ) ` M ) as "p(M)" in the comments. The characteristric matrix ( ( X .x. .1. ) .- ( T ` M ) ) will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix M must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 19666), also the characteristic polynomial can be written in this way:

p(M) = sumi ( pi * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring R, and the coefficients are elements of the ring R.

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 20697. Here, * is the scalar multiplication in the algebra of the matrices over the ring R.

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "AB") are used:

class variable/ informal identifier definiens explanation
N An arbitrary finite set, used as dimension for matrices
R An arbitrary (commutative) ring: R e. CRing
B(R) ( Base ` R ) Base set of (the ring) R
A ( N Mat R ) Algebra of N x N matrices over (the ring) R
B ( Base ` A ) Base set of the algebra of N x N matrices, i .e. the set of all N x N matrices
M An arbitrary N x N matrix
P (Poly1 ` R ) The algebra of polynomials over (the ring) R
B(P) ( Base ` P ) Base set of the algebra of polynomials, i .e. the set of all polynomials
X, XR (var1 ` R ) The variable of polynomials over (the ring) R
Y, XA (var1 ` A ) The variable of polynomials over matrices of the algebra A
.^ (.g ` (mulGrp ` P ) ) The group exponentiation for polynomials over (the ring) R
^ Arbitrary group exponentiation
U (algSc ` P ) The injection of scalars, i.e. elements of (the ring) R into the base set of the algebra of polynomials over R
( U ` p ), S(p) ( (algSc ` P ) ` p ) The element p of (the ring) R represented as polynomial over R
Y ( N Mat P ) Algebra of N x N matrices over (the polynomial ring) P over the ring R
B(Y) ( Base ` Y ) Base set of the algebra of polynomial N x N matrices, i .e. the set of all polynomial N x N matrices
Q (Poly1 ` A ) Algebra of polynomials over the ring of N x N matrices over the ring R
B(Q) ( Base ` Q ) Base set of the algebra of polynomials over the ring of N x N matrices over the ring R, i .e. the set of all polynomials having N x N matrices as coefficients
.+, + ( +g ` Y ) The addition of polynomial matrices
.-, - ( -g ` Y ) The subtraction of polynomial matrices
.x., *Y ( .s ` Y ) The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A ( .s ` A ) The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q ( .s ` Q ) The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
.X., xY ( .r ` Y ) The multiplication of polynomial matrices
xA ( .r ` A ) The multiplication of matrices
xQ ( .r ` Q ) The multiplication of polynomials over matrices
.1., 1Y ( 1r ` Y ) The identity matrix in the algebra of polynomial matrices over R
1A ( 1r ` A ) The identity matrix in the algebra of matrices over R
.0., 0Y ( 0g ` Y ) The zero matrix in the algebra of matrices consisting of polynomials
T ( N matToPolyMat R ) The transformation of an N x N matrix over R into a polynomial N x N matrix over R
T1(M) ( T ` M ) The matrix M transformed into a polynomial N x N matrix over R
U(M) ( U ` M ) The (constant) polynomial N x N matrix M transformed into a matrix over the ring R. Inverse function of T: ( T ` ( U ` M ) ) = M (see m2cpminvid2 20560 )
T2(M) ( ( N pMatToMatPoly R ) ` M ) The polynomial N x N matrix M transformed into a polynomial over the N x N matrices over R
I, c(M) ( ( X .x. .1. ) .- ( T ` M ) ) The characteristic matrix of a matrix M, i.e. the matrix whose determinant is the characteristic polynomial of the matrix M
C ( N CharPlyMat R ) The function mapping a matrix over (a ring) R to its characteristic polynomial
K, p(M) ( C ` M ) The characteristic polynomial of a matrix over (a ring) R
H ( K .x. .1. ) The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
J ( N maAdju P ) The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
W, adj(cm(M)) ( J ` I ) The adjugate of the characteristic matrix of the matrix M


Using this notation, we have:
  1. "c(M) e. B(Y)", or I e. ( Base ` Y ), see chmatcl 20633
  2. "p(M) e. B(P)", or K e. ( Base ` P ), see chpmatply1 20637
  3. "T(M) e. B(Y)", or ( T ` M ) e. ( Base ` Y ), see mat2pmatbas 20531
  4. J : ( Base ` Y ) --> ( Base ` Y ), see maduf 20447
  5. "adj(cm(M)) e. B(Y)", or W e. ( Base ` Y )


Following the proof shown in Wikipedia, the following steps are performed:
  1. Write W, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 20593:
    adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
    where b(i) are matrices over the ring R, so T1(b(i)) are constant polynomial matrices.
    This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write W as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of ( I .X. W ) as infinite, but finitely supported sum, see step 3.
  2. Write ( I .X. W ), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 20682. This representation is obtained by replacing W by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
    cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
    where b(i) are matrices over R, so T1(b(i)) are constant polynomial matrices:
    cm(M) *Y adj(cm(M))
    = cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 20593 (step 1.)]
    = ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
    = ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 18600]
    = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 20681]
    This step corresponds partially to (4) in Wikipedia.
  3. Write ( I .X. W ) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 20683:
    cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
    This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 20659, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 20660). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
  4. Write H = ( K .x. .1. ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 20673:
    p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
    The proof of cpmidgsum 20673 is making use of pmatcollpwscmat 20596, because H = ( K .x. .1. ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring R, S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
  5. Transform the sum representation of ( I .X. W ) from step 3. into polynomials over matrices:
    T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 20688]
    where U(G(i)) is a matrix over the ring R.
  6. Transform the sum representation of H from step 4. into polynomials over matrices:
    T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 20678]
  7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 20672 to obtain the equation
    sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
    sumi ( U(G(i)) *Q XA ^i )
    = T2(cm(M) * adj(cm(M))) [see step 5.]
    = T2(p(m) *Y IY) [see cpmadurid 20672]
    = sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
    Note that this step is contained in the proof of chcoeffeq 20691, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
  8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 20690. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
  9. Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 20691:
    sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
    This corresponds to (7) in Wikipedia.
  10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 20692:
    sumi ( pi *A M^i )
    = sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 20689]
    = sumi ( M^i xA U(G(i)) ) [see chcoeffeq 20691]
  11. Apply the theorem for telescoping sums, see telgsumfz 18387, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
    sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 20671:
    sumi ( T1(M)^i xY G(i) )
    = sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
    + ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 20670]
    = ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 18387]
    = 0Y [see grpnpncan0 17511] This step corresponds partially to (8) in Wikipedia.
  12. Since T is a ring homomorphism (see mat2pmatrhm 20539), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 20693:
    sumi ( pi *A M^i )
    = sumi ( M^i xA U(G(i)) ) [see cayhamlem3 20692 (step 10.)]
    = U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 20558]
    = U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 18340]
    = U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 18727]
    = U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 17583]
    = U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 20560 ]
  13. Finally, combine the results of steps 11. and 12., and use the fact that T (and therefore also its inverse U) is an injective ring homomorphism (see mat2pmatf1 20534 and mat2pmatrhm 20539) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 20695 resp. cayleyhamilton0 20694:
    sumi ( pi *A M^i )
    = U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 20693 (step 12.)]
    = U(0Y ) [see cayhamlem1 20671 (step 11.)]
    = 0A [see m2cpminv0 20566]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring R and its representation as matrix over the polynomial ring over the ring R in general!
 
Theoremcpmadurid 20672 The right-hand fundamental relation of the adjugate (see madurid 20450) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- C = ( N CharPlyMat R )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- T = ( N matToPolyMat R )   &    |- .- = ( -g ` Y )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- I = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- .X. = ( .r ` Y )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( I .X. ( J ` I ) ) = ( ( C ` M ) .x. .1. ) )
 
Theoremcpmidgsum 20673* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> H = ( Y gsumg ( n e. NN0 |-> ( ( n .^ X ) .x. ( ( U ` ( (coe1 ` K ) ` n ) ) .x. .1. ) ) ) ) )
 
Theoremcpmidgsumm2pm 20674* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   &    |- O = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- T = ( N matToPolyMat R )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> H = ( Y gsumg ( n e. NN0 |-> ( ( n .^ X ) .x. ( T ` ( ( (coe1 ` K ) ` n ) .* O ) ) ) ) ) )
 
Theoremcpmidpmatlem1 20675* Lemma 1 for cpmidpmat 20678. (Contributed by AV, 13-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   &    |- O = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- T = ( N matToPolyMat R )   &    |- G = ( k e. NN0 |-> ( ( (coe1 ` K ) ` k ) .* O ) )   =>    |- ( L e. NN0 -> ( G ` L ) = ( ( (coe1 ` K ) ` L ) .* O ) )
 
Theoremcpmidpmatlem2 20676* Lemma 2 for cpmidpmat 20678. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   &    |- O = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- T = ( N matToPolyMat R )   &    |- G = ( k e. NN0 |-> ( ( (coe1 ` K ) ` k ) .* O ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> G e. ( B ^m NN0 ) )
 
Theoremcpmidpmatlem3 20677* Lemma 3 for cpmidpmat 20678. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   &    |- O = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- T = ( N matToPolyMat R )   &    |- G = ( k e. NN0 |-> ( ( (coe1 ` K ) ` k ) .* O ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> G finSupp ( 0g ` A ) )
 
Theoremcpmidpmat 20678* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- U = (algSc ` P )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   &    |- O = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- T = ( N matToPolyMat R )   &    |- W = ( Base ` Y )   &    |- Q = (Poly1 ` A )   &    |- Z = (var1 ` A )   &    |- .xb = ( .s ` Q )   &    |- E = (.g ` (mulGrp ` Q ) )   &    |- I = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( I ` H ) = ( Q gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* O ) .xb ( n E Z ) ) ) ) )
 
TheoremcpmadugsumlemB 20679* Lemma B for cpmadugsum 20683. (Contributed by AV, 2-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN0 /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( X .x. .1. ) .X. ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) ) = ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( ( i + 1 ) .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) )
 
TheoremcpmadugsumlemC 20680* Lemma C for cpmadugsum 20683. (Contributed by AV, 2-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN0 /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( T ` M ) .X. ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) ) = ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( ( T ` M ) .X. ( T ` ( b ` i ) ) ) ) ) ) )
 
TheoremcpmadugsumlemF 20681* Lemma F for cpmadugsum 20683. (Contributed by AV, 7-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   &    |- .+ = ( +g ` Y )   &    |- .- = ( -g ` Y )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( ( X .x. .1. ) .X. ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) ) .- ( ( T ` M ) .X. ( Y gsumg ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) ) ) = ( ( Y gsumg ( i e. ( 1 ... s ) |-> ( ( i .^ X ) .x. ( ( T ` ( b ` ( i - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` i ) ) ) ) ) ) ) .+ ( ( ( ( s + 1 ) .^ X ) .x. ( T ` ( b ` s ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) ) )
 
Theoremcpmadugsumfi 20682* The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   &    |- .+ = ( +g ` Y )   &    |- .- = ( -g ` Y )   &    |- I = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( I .X. ( J ` I ) ) = ( ( Y gsumg ( i e. ( 1 ... s ) |-> ( ( i .^ X ) .x. ( ( T ` ( b ` ( i - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` i ) ) ) ) ) ) ) .+ ( ( ( ( s + 1 ) .^ X ) .x. ( T ` ( b ` s ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) ) )
 
Theoremcpmadugsum 20683* The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   &    |- .+ = ( +g ` Y )   &    |- .- = ( -g ` Y )   &    |- I = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( I .X. ( J ` I ) ) = ( Y gsumg ( i e. NN0 |-> ( ( i .^ X ) .x. ( G ` i ) ) ) ) )
 
Theoremcpmidgsum2 20684* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   &    |- .+ = ( +g ` Y )   &    |- .- = ( -g ` Y )   &    |- I = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- H = ( K .x. .1. )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) H = ( Y gsumg ( i e. NN0 |-> ( ( i .^ X ) .x. ( G ` i ) ) ) ) )
 
Theoremcpmidg2sum 20685* Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- X = (var1 ` R )   &    |- .^ = (.g ` (mulGrp ` P ) )   &    |- .x. = ( .s ` Y )   &    |- .X. = ( .r ` Y )   &    |- .1. = ( 1r ` Y )   &    |- .+ = ( +g ` Y )   &    |- .- = ( -g ` Y )   &    |- I = ( ( X .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- U = (algSc ` P )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( Y gsumg ( i e. NN0 |-> ( ( i .^ X ) .x. ( ( U ` ( (coe1 ` K ) ` i ) ) .x. .1. ) ) ) ) = ( Y gsumg ( i e. NN0 |-> ( ( i .^ X ) .x. ( G ` i ) ) ) ) )
 
Theoremcpmadumatpolylem1 20686* Lemma 1 for cpmadumatpoly 20688. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- S = ( N ConstPolyMat R )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- Z = (var1 ` R )   &    |- D = ( ( Z .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- W = ( Base ` Y )   &    |- Q = (Poly1 ` A )   &    |- X = (var1 ` A )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- U = ( N cPolyMatToMat R )   =>    |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( U o. G ) e. ( B ^m NN0 ) )
 
Theoremcpmadumatpolylem2 20687* Lemma 2 for cpmadumatpoly 20688. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- S = ( N ConstPolyMat R )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- Z = (var1 ` R )   &    |- D = ( ( Z .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- W = ( Base ` Y )   &    |- Q = (Poly1 ` A )   &    |- X = (var1 ` A )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- U = ( N cPolyMatToMat R )   =>    |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( U o. G ) finSupp ( 0g ` A ) )
 
Theoremcpmadumatpoly 20688* The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- T = ( N matToPolyMat R )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- S = ( N ConstPolyMat R )   &    |- .x. = ( .s ` Y )   &    |- .1. = ( 1r ` Y )   &    |- Z = (var1 ` R )   &    |- D = ( ( Z .x. .1. ) .- ( T ` M ) )   &    |- J = ( N maAdju P )   &    |- W = ( Base ` Y )   &    |- Q = (Poly1 ` A )   &    |- X = (var1 ` A )   &    |- .* = ( .s ` Q )   &    |- .^ = (.g ` (mulGrp ` Q ) )   &    |- U = ( N cPolyMatToMat R )   &    |- I = ( N pMatToMatPoly R )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( I ` ( D .X. ( J ` D ) ) ) = ( Q gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) .* ( n .^ X ) ) ) ) )
 
Theoremcayhamlem2 20689 Lemma for cayhamlem3 20692. (Contributed by AV, 24-Nov-2019.)
|- K = ( Base ` R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- .^ = (.g ` (mulGrp ` A ) )   &    |- .x. = ( .r ` A )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( H e. ( K ^m NN0 ) /\ L e. NN0 ) ) -> ( ( H ` L ) .* ( L .^ M ) ) = ( ( L .^ M ) .x. ( ( H ` L ) .* .1. ) ) )
 
Theoremchcoeffeqlem 20690* Lemma for chcoeffeq 20691. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- W = ( Base ` Y )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- U = ( N cPolyMatToMat R )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
 
Theoremchcoeffeq 20691* The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- W = ( Base ` Y )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- U = ( N cPolyMatToMat R )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) )
 
Theoremcayhamlem3 20692* Lemma for cayhamlem4 20693. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- W = ( Base ` Y )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- U = ( N cPolyMatToMat R )   &    |- .^ = (.g ` (mulGrp ` A ) )   &    |- .x. = ( .r ` A )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( A gsumg ( n e. NN0 |-> ( ( (coe1 ` K ) ` n ) .* ( n .^ M ) ) ) ) = ( A gsumg ( n e. NN0 |-> ( ( n .^ M ) .x. ( U ` ( G ` n ) ) ) ) ) )
 
Theoremcayhamlem4 20693* Lemma for cayleyhamilton 20695. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- .0. = ( 0g ` Y )   &    |- T = ( N matToPolyMat R )   &    |- C = ( N CharPlyMat R )   &    |- K = ( C ` M )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- W = ( Base ` Y )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- U = ( N cPolyMatToMat R )   &    |- .^ = (.g ` (mulGrp ` A ) )   &    |- E = (.g ` (mulGrp ` Y ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( A gsumg ( n e. NN0 |-> ( ( (coe1 ` K ) ` n ) .* ( n .^ M ) ) ) ) = ( U ` ( Y gsumg ( n e. NN0 |-> ( ( n E ( T ` M ) ) .X. ( G ` n ) ) ) ) ) )
 
Theoremcayleyhamilton0 20694* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 20695 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 20696)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` A )   &    |- .1. = ( 1r ` A )   &    |- .* = ( .s ` A )   &    |- .^ = (.g ` (mulGrp ` A ) )   &    |- C = ( N CharPlyMat R )   &    |- K = (coe1 ` ( C ` M ) )   &    |- P = (Poly1 ` R )   &    |- Y = ( N Mat P )   &    |- .X. = ( .r ` Y )   &    |- .- = ( -g ` Y )   &    |- Z = ( 0g ` Y )   &    |- W = ( Base ` Y )   &    |- E = (.g ` (mulGrp ` Y ) )   &    |- T = ( N matToPolyMat R )   &    |- G = ( n e. NN0 |-> if ( n = 0 , ( Z .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , Z , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )   &    |- U = ( N cPolyMatToMat R )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. )
 
Theoremcayleyhamilton 20695* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` A )   &    |- C = ( N CharPlyMat R )   &    |- K = (coe1 ` ( C ` M ) )   &    |- .* = ( .s ` A )   &    |- .^ = (.g ` (mulGrp ` A ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. )
 
TheoremcayleyhamiltonALT 20696* Alternate proof of cayleyhamilton 20695, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20694 directly, but has the same structure as the proof of cayleyhamilton0 20694. In contrast to the proof of cayleyhamilton0 20694, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` A )   &    |- C = ( N CharPlyMat R )   &    |- K = (coe1 ` ( C ` M ) )   &    |- .* = ( .s ` A )   &    |- .^ = (.g ` (mulGrp ` A ) )   =>    |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. )
 
Theoremcayleyhamilton1 20697* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 20695, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients ( F ` n ), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` A )   &    |- C = ( N CharPlyMat R )   &    |- K = (coe1 ` ( C ` M ) )   &    |- .* = ( .s ` A )   &    |- .^ = (.g ` (mulGrp ` A ) )   &    |- L = ( Base ` R )   &    |- X = (var1 ` R )   &    |- P = (Poly1 ` R )   &    |- .x. = ( .s ` P )   &    |- E = (.g ` (mulGrp ` P ) )   &    |- Z = ( 0g ` R )   =>    |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) -> ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) = .0. ) )
 
PART 12  BASIC TOPOLOGY
 
12.1  Topology
 
12.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 20719), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 20699), then the function which associates with a set the set of topologies on it (df-topon 20716), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 20737). Of course, a topology is the same thing as a topology on a set (see toprntopon 20729).

 
12.1.1.1  Topologies
 
Syntaxctop 20698 Syntax for the class of topologies.
class Top
 
Definitiondf-top 20699* Define the class of topologies. It is a proper class (see topnex 20800). See istopg 20700 and istop2g 20701 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
 
Theoremistopg 20700* Express the predicate " J is a topology." See istop2g 20701 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
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