QR factorization

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QR factorization A form of matrix factorization widely used in numerical linear algebra. For A, an m × n, mn, real square matrix, the factorization takes the form A = QR

where Q is an m × m orthogonal matrix and R is an m × n matrix whose first n rows form an upper (or right) triangular matrix. An important application is in solving overdetermined linear systems of equations of the form Ax = b, m > n; b is an m-component column vector and x is a column vector of n unknowns. The QR factorization, under appropriate conditions, reduces the problem to solving a simpler square upper triangular system of the form Rx = c.

For a square matrix, m = n, a further major application is in computing the eigenvalues and eigenvectors of A. Here a sequence of QR factorizations are carried out in an iteration scheme that ultimately reduces A to a matrix of a particularly simple form whose eigenvalues are the same as those of A. The eigenvalues (and if required, eigenvectors) can now be easily computed.