User`s guide

QR Factorization
5-349
5QR Factorization
Purpose Factor a rectangular matrix into unitary and upper triangular components.
Library Math Functions / Matrices and Linear Algebra / Matrix Factorizations
Description The QR Factorization block uses modified Gram-Schmidt iteration to factor a
column permutation of the M-by-N input matrix A as
where Q is an M-by-min(M,N) unitary matrix, and R is a min(M,N)-by-N
upper-triangular matrix. A length-M vector input is treated as an M-by-1
matrix, and is always sample-based.
The column-pivoted matrix A
e
contains the columns of A permuted as
indicated by the contents of length-N permutation vector E.
Ae = A(:,E) % Equivalent MATLAB code
The block selects a column permutation vector E, which ensures that the
diagonal elements of matrix R are arranged in order of decreasing magnitude.
QR factorization is an important tool for solving linear systems of equations
because of good error propagation properties and the invertability of unitary
matrices.
Unlike LU and Cholesky factorizations, the matrix A does not need to be
square for QR factorization. Note, however, that QR factorization requires
twice as many operations as Gaussian elimination.
Example A sample factorization is shown below. The input to the block is matrix A,
which is permuted according to vector E to produce matrix A
e
. Matrix A
e
is
factored to produce the Q and R output matrices.
A
e
QR=
r
i 1 j 1+,+
r
ij,
ij=>
Q
1–
Q
*
=