Two-sided hyperbolic SVD

In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i e., A = U Sigma V(vertical bar*vertical bar), where U and V(vertical bar*vertical bar) are J-unitary (J = diag(+/-1)) and E is a real diagonal matrix of "double-hyperbolic" singular values. We show that, with some natural conditions, such decomposition exists without the use of hyper-exchange matrices. In other words, U and V(vertical bar*vertical bar) are really J-unitary with regard to J and not some matrix (J) over cap which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition. (C) 2010 Elsevier Inc. All rights reserved.