Generalized Hyperbolic Singular Value Decompositions And Applications In Generalized Inverse And Indefinite Least Squares Problems

In this paper, firstly, we separately give the CS decomposition of strong-unitary matrices and the hyperbolic CS decomposition of strong-substandard J-orthogonal matrices by the CS decomposition of orthogonal ma-trices and the hyperbolic CS decomposition of J-orthogonal matrices,and study properties of J-orthogonal matrices;Secondly we obtain several gener-alized hyperbolic singular values decompositions theorems:hyperbolic correl-ative criterion decomposition(HCCD),hyperbolic singular values decomposi-tionsof product(P-HSVD) and hyperbolic singular values decompositions of quotient(Q-HSVD) in the foundations of hyperbolic singular value HSVD and hy-perbolic CS decomposition of J-orthogonal matrices(To Q-HSVD decompositions ,there is statement in reference [8],but the conclusion in this paper is different from it.In the posterior preface,we give detailed statements to their difference.) fur-thermore,we compute product matrix (AB){1},(AB){2} and (AB){1,2} using P-HSVD and HCCD,and solve the non-constrained indefinite least squares prob-lems and the equality constrained indefinite least squares problems by Q-HSVD method.Otherwise, when we adopt Q-HSVD method to solve the equality constrained indefinite least squares problems,we assume matrix AH A-BH B is non-singular .If matrix AHA-BHB is non-singular,we shall use Q-HSVD method to solve the equality constrained indefinite least squares problems;if matrix AH A-BHB is singular,we will employ Q-SVD method to solve the equality constrained indefinite least squares problems.We also employ Q-HSVD method to solve the indefinite least squares prob-lems,furthermore,this method is very easy.Last, we do corresponding numerical experiment,and they demonstrate the availability of these algorithms.