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:

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.

*QR*FactorizationFactorization*LQ*Factorization with Column Pivoting*QR*- Complete Orthogonal Factorization
- Other Factorizations