Orthogonal Factorizations and Linear Least Squares Problems

LAPACK provides a number of routines for factorizing a general rectangular m-by-n matrix A, as the product of an orthogonal matrix (unitary if complex) and a triangular (or possibly trapezoidal) matrix.

A real matrix Q is orthogonal if Q^T Q = I; a complex matrix Q is unitary if Q^H Q = I. Orthogonal or unitary matrices have the important property that they leave the two-norm of a vector invariant:

$$\Vert x\vert\vert _2 = \Vert Qx\Vert _2, \quad \mbox{if $Q$\ is orthogonal or unitary.}$$

As a result, they help to maintain numerical stability because they do not amplify rounding errors.

Orthogonal factorizations are used in the solution of linear least squares problems. They may also be used to perform preliminary steps in the solution of eigenvalue or singular value problems.