On a variational formulation of the QSVD and the RSVD

Abstract

Recently, M.T. Chu, R.F. Funderlic and G.H. Golub [SIAM J. Matrix Anal. Appl., 18 (1997) 1082–1092] presented a variational formulation for the quotient singular value decomposition (QSVD) of two matrices A∈ R n×m,C∈ R p×m which is a generalization to two matrices of the ordinary singular value decomposition (SVD) and characterizes the role of two orthogonal matrices in QSVD. In this paper, we give an alternative derivation of this variational formulation and extend it to establish an analogous variational formulation for the Restricted Singular Value Decomposition (RSVD) of a matrix triplet A∈ R n×m,B∈ R n×l,C∈ R p×m , which provides a new understanding of the orthogonal matrices appearing in this decomposition.