qr — QR decomposition
[Q,R]=qr(X [,"e"]) [Q,R,E]=qr(X [,"e"]) [Q,R,rk,E]=qr(X [,tol])
real or complex matrix
nonnegative real number
square orthogonal or unitary matrix
matrix with same dimensions as X
permutation matrix
integer (QR-rank of X)
produces an upper triangular matrix R of the same dimension as X and an orthogonal (unitary in the complex case) matrix Q so that X = Q*R. [Q,R] = qr(X,"e") produces an "economy size": If X is m-by-n with m > n, then only the first n columns of Q are computed as well as the first n rows of R.
From Q*R = X , it follows that
the kth column of the matrix X, is expressed as a linear combination
of the k first columns of Q (with coefficients R(1,k), ..., R(k,k) ). The k first columns of Q make an orthogonal basis
of the subspace spanned by the k first comumns of X. If column k
of X (i.e. X(:,k) ) is a linear combination of the first
p columns of X, then the entries R(p+1,k), ..., R(k,k)
are zero. It this situation, R is upper trapezoidal. If X has
rank rk, rows R(rk+1,:), R(rk+2,:), ... are zeros.
produces a (column) permutation matrix E, an upper
triangular R with decreasing diagonal elements and an
orthogonal (or unitary) Q so that X*E = Q*R.
If rk is the rank of X, the
rk first entries along the main diagonal of
R, i.e. R(1,1), R(2,2), ..., R(rk,rk)
are all different from zero. [Q,R,E] = qr(X,"e")
produces an "economy size":
If X is m-by-n with m > n, then only the first n
columns of Q are computed as well as the first n
rows of R.
returns rk = rank estimate of X i.e. rk is the number of diagonal elements in R which are larger than a given threshold tol.
returns rk = rank estimate of X
i.e. rk is the number of diagonal elements in
R which are larger than
tol=R(1,1)*%eps*max(size(R)). See rankqr
for a rank revealing QR factorization, using the condition number
of R.
// QR factorization, generic case // X is tall (full rank) X=rand(5,2);[Q,R]=qr(X); [Q'*X R] //X is fat (full rank) X=rand(2,3);[Q,R]=qr(X); [Q'*X R] //Column 4 of X is a linear combination of columns 1 and 2: X=rand(8,5);X(:,4)=X(:,1)+X(:,2); [Q,R]=qr(X); R, R(:,4) //X has rank 2, rows 3 to $ of R are zero: X=rand(8,2)*rand(2,5);[Q,R]=qr(X); R //Evaluating the rank rk: column pivoting ==> rk first //diagonal entries of R are non zero : A=rand(5,2)*rand(2,5); [Q,R,rk,E] = qr(A,1.d-10); norm(Q'*A-R) svd([A,Q(:,1:rk)]) //span(A) =span(Q(:,1:rk))