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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | grpvlinv 20201 | Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Theorem | grpvrinv 20202 | Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.) |
Theorem | mhmvlin 20203 | Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
MndHom | ||
Theorem | ringvcl 20204 | Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Theorem | gsumcom3 20205* | A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.) |
CMnd g g g g | ||
Theorem | gsumcom3fi 20206* | A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
CMnd g g g g | ||
Theorem | mamucl 20207 | Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
maMul | ||
Theorem | mamuass 20208 | Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
maMul maMul maMul maMul | ||
Theorem | mamudi 20209 | Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
maMul | ||
Theorem | mamudir 20210 | Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
maMul | ||
Theorem | mamuvs1 20211 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
maMul | ||
Theorem | mamuvs2 20212 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
maMul | ||
In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that Mat is a left module, see matlmod 20235. That Mat is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, Mat is called "matrix ring" or "matrix algebra" already in this subsection. | ||
Syntax | cmat 20213 | Syntax for the square matrix algebra. |
Mat | ||
Definition | df-mat 20214* | Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.) |
Mat freeLMod sSet maMul | ||
Theorem | matbas0pc 20215 | There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.) |
Mat | ||
Theorem | matbas0 20216 | There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.) |
Mat | ||
Theorem | matval 20217 | Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod maMul sSet | ||
Theorem | matrcl 20218 | Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat | ||
Theorem | matbas 20219 | The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod | ||
Theorem | matplusg 20220 | The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod | ||
Theorem | matsca 20221 | The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod Scalar Scalar | ||
Theorem | matvsca 20222 | The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod | ||
Theorem | mat0 20223 | The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod | ||
Theorem | matinvg 20224 | The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat freeLMod | ||
Theorem | mat0op 20225* | Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.) |
Mat | ||
Theorem | matsca2 20226 | The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat Scalar | ||
Theorem | matbas2 20227 | The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.) |
Mat | ||
Theorem | matbas2i 20228 | A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.) |
Mat | ||
Theorem | matbas2d 20229* | The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.) |
Mat | ||
Theorem | eqmat 20230* | Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.) |
Mat | ||
Theorem | matecl 20231 | Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.) |
Mat | ||
Theorem | matecld 20232 | Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.) |
Mat | ||
Theorem | matplusg2 20233 | Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat | ||
Theorem | matvsca2 20234 | Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat | ||
Theorem | matlmod 20235 | The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat | ||
Theorem | matgrp 20236 | The matrix ring is a group. (Contributed by AV, 21-Dec-2019.) |
Mat | ||
Theorem | matvscl 20237 | Closure of the scalar multiplication in the matrix ring. (lmodvscl 18880 analog.) (Contributed by AV, 27-Nov-2019.) |
Mat | ||
Theorem | matsubg 20238 | The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.) |
Mat freeLMod | ||
Theorem | matplusgcell 20239 | Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
Mat | ||
Theorem | matsubgcell 20240 | Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
Mat | ||
Theorem | matinvgcell 20241 | Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.) |
Mat | ||
Theorem | matvscacell 20242 | Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.) |
Mat | ||
Theorem | matgsum 20243* | Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.) |
Mat finSupp g g | ||
The main result of this subsection are the theorems showing that Mat is a ring (see matring 20249) and an associative algebra (see matassa 20250). Additionally, theorems for the identity matrix and transposed matrices are provided. | ||
Theorem | matmulr 20244 | Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Mat maMul | ||
Theorem | mamumat1cl 20245* | The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
Theorem | mat1comp 20246* | The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.) |
Theorem | mamulid 20247* | The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
maMul | ||
Theorem | mamurid 20248* | The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
maMul | ||
Theorem | matring 20249 | Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat | ||
Theorem | matassa 20250 | Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Mat AssAlg | ||
Theorem | matmulcell 20251* | Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) |
Mat g | ||
Theorem | mpt2matmul 20252* | Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.) |
Mat g | ||
Theorem | mat1 20253* | Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.) |
Mat | ||
Theorem | mat1ov 20254 | Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.) |
Mat | ||
Theorem | mat1bas 20255 | The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.) |
Mat Mat | ||
Theorem | matsc 20256* | The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.) |
Mat | ||
Theorem | ofco2 20257 | Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Theorem | oftpos 20258 | The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
tpos tpos tpos | ||
Theorem | mattposcl 20259 | The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.) |
Mat tpos | ||
Theorem | mattpostpos 20260 | The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.) |
Mat tpos tpos | ||
Theorem | mattposvs 20261 | The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Mat tpos tpos | ||
Theorem | mattpos1 20262 | The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Mat tpos | ||
Theorem | tposmap 20263 | The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.) |
tpos | ||
Theorem | mamutpos 20264 | Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.) |
maMul maMul tpos tpos tpos | ||
Theorem | mattposm 20265 | Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Mat tpos tpos tpos | ||
Theorem | matgsumcl 20266* | Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.) |
Mat mulGrp g | ||
Theorem | madetsumid 20267* | The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) |
Mat mulGrp RHom pmSgn g g | ||
Theorem | matepmcl 20268* | Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.) |
Mat | ||
Theorem | matepm2cl 20269* | Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.) |
Mat | ||
Theorem | madetsmelbas 20270* | A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.) |
pmSgn RHom Mat mulGrp g | ||
Theorem | madetsmelbas2 20271* | A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.) |
pmSgn RHom Mat mulGrp g | ||
As already mentioned before, and shown in mat0dimbas0 20272, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 20276. For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 20293. | ||
Theorem | mat0dimbas0 20272 | The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.) |
Mat | ||
Theorem | mat0dim0 20273 | The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Mat | ||
Theorem | mat0dimid 20274 | The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Mat | ||
Theorem | mat0dimscm 20275 | The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Mat | ||
Theorem | mat0dimcrng 20276 | The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.) |
Mat | ||
Theorem | mat1dimelbas 20277* | A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.) |
Mat | ||
Theorem | mat1dimbas 20278 | A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.) |
Mat | ||
Theorem | mat1dim0 20279 | The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.) |
Mat | ||
Theorem | mat1dimid 20280 | The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.) |
Mat | ||
Theorem | mat1dimscm 20281 | The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) |
Mat | ||
Theorem | mat1dimmul 20282 | The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.) |
Mat | ||
Theorem | mat1dimcrng 20283 | The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.) |
Mat | ||
Theorem | mat1f1o 20284* | There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1rhmval 20285* | The value of the ring homomorphism . (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1rhmelval 20286* | The value of the ring homomorphism . (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1rhmcl 20287* | The value of the ring homomorphism is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1f 20288* | There is a function from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1ghm 20289* | There is a group homomorphism from the additive group of a ring to the additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat | ||
Theorem | mat1mhm 20290* | There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat mulGrp mulGrp MndHom | ||
Theorem | mat1rhm 20291* | There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat RingHom | ||
Theorem | mat1rngiso 20292* | There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Mat RingIso | ||
Theorem | mat1ric 20293 | A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.) |
Mat | ||
According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple of the identity matrix . Its effect on a vector is scalar multiplication by [see scmatscm 20319!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name. The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat 20296 and df-scmat 20297), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20307), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20326), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20329) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20327). | ||
Syntax | cdmat 20294 | Extend class notation for the algebra of diagonal matrices. |
DMat | ||
Syntax | cscmat 20295 | Extend class notation for the algebra of scalar matrices. |
ScMat | ||
Definition | df-dmat 20296* | Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.) |
DMat Mat | ||
Definition | df-scmat 20297* | Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn". (Contributed by AV, 8-Dec-2019.) |
ScMat Mat | ||
Theorem | dmatval 20298* | The set of x diagonal matrices over (a ring) . (Contributed by AV, 8-Dec-2019.) |
Mat DMat | ||
Theorem | dmatel 20299* | A x diagonal matrix over (a ring) . (Contributed by AV, 18-Dec-2019.) |
Mat DMat | ||
Theorem | dmatmat 20300 | An x diagonal matrix over (the ring) is an x matrix over (the ring) . (Contributed by AV, 18-Dec-2019.) |
Mat DMat |
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