Towards a Generalized Singular Value Decomposition

We suggest a form for, and give a constructive derivation of, the generalized singular value decomposition of any two matrices having the same number of columns. We outline its desirable characteristics and compare it to an earlier suggestion by Van Loan SIAM J. Numer. Anal., 13 (1976), pp. 76–83. The present form largely follows from the work of Van Loan, but is slightly more general and computationally more amenable than that in the paper cited. We also prove a useful extension of a theorem of Stewart SIAM Rev. 19 (1977), pp. 634–662 on unitary decompositions of submatrices of a unitary matrix.