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### LQ Factorization

The LQ factorization is given by

where L is m-by-m lower triangular, Q is n-by-n orthogonal (or unitary), Q1 consists of the first m rows of Q, and Q2 the remaining n-m rows.

This factorization is computed by the routine xGELQF, and again Q is represented as a product of elementary reflectors; xORGLQ (or xUNGLQ in the complex case) can generate all or part of Q, and xORMLQ (or xUNMLQ ) can pre- or post-multiply a given matrix by Q or QT (QH if Q is complex).

The LQ factorization of A is essentially the same as the QR factorization of AT (AH if A is complex), since

The LQ factorization may be used to find a minimum norm solution of an underdetermined system of linear equations A x = b where A is m-by-n with m < n and has rank m. The solution is given by

and may be computed by calls to xTRTRS and xORMLQ.

Next: QR Factorization with Column Up: Orthogonal Factorizations and Linear Previous: QR Factorization Contents Index