A Continuous Approach for the Computation of the Hyperbolic Singular Value Decomposition

In this paper a continuous approach based on the Projected Gradient Flow technique is presented in order to find a generalization of the Singular Value Decomposition (SVD) of a rectangular matrix called Hyperbolic SVD. If A is a m× n real matrix with full column rank and if G is a n× n diagonal sign matrix, i.e. g ii =± 1, the Hyperbolic Singular Value Decomposition of the pair (A,G) is defined as A=UΣV, where U is orthogonal, Σ is diagonal with positive entries and V is hypernormal (or G-orthogonal), i.e. V T GV=G. In this work we use a continuous approach based on the projected gradient technique obtaining two differential systems, the first one evolving on group of orthogonal matrices and the second on the quadratic group related to G. A numerical test is reported in order to show the effectiveness of the approach.