Recently, M.T. Chu, R.F, Funderlic and G.H. Golub [SIAM J, Matrix Anal, App
l., 18 (1997) 1081-1092] presented a variational formulation for the quotie
nt singular value decomposition (QSVD) of two matrices A is an element of R
-nxm, C is an element of R-pxm 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 an
alogous variational formulation for the Restricted Singular Value Decomposi
tion (RSVD) of a matrix triplet A is an element of R-nxm, B is an element o
f R-nxl, C is an element of R-pxm, which provides a new understanding of th
e orthogonal matrices appearing in this decomposition. (C) 2000 Elsevier Sc
ience Inc. hll rights reserved. AMS classification: 65F15; 65H15.