The generalized RQ (GRQ) factorization of an m-by-n matrix A and
a p-by-n matrix B is given by the pair of factorizations
where Q and Z are respectively n-by-n and p-by-p orthogonal matrices (or unitary matrices if A and B are complex). R has the form
where or is upper triangular. T has the form
where is upper triangular.
Note that if B is square and nonsingular, the GRQ factorization of
A and B implicitly gives the RQ factorization of the matrix :
without explicitly computing the matrix inverse or the product .
The routine PxGGRQF computes the GRQ factorization by computing first the RQ factorization of A and then the QR factorization of . The orthogonal (or unitary) matrices Q and Z can be formed explicitly or can be used just to multiply another given matrix in the same way as the orthogonal (or unitary) matrix in the RQ factorization (see section 3.3.2).