Orthoghonal Factorizations

OpenBLAS 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 QT Q = I ; a complex matrix Q is unitary if QH Q = I . Orthogonal or unitary matrices have the important property that they leave the two-norm of a vector invariant:

||x||2 = ||Qx||2, 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.