The Application Of Matrix Decomposition In The Matrix Perturbation Theory

The matrix decomposition plays very important role in matrix theory,especially in the study of perturbation analysis of matrices.The main tool in this paper is singular value decomposition and (generalized) C-S decomposition.By applying these two tools,we mainly discuss the problems as follows:1) Perturbation identities for the generalized polar decomposition:By applying singular value decomposition, we study the perturbation bounds of the generalized polar decomposition under unitary invariant norms when A is perturbed by addition or multiplication,respectively and give our bounds in identities.2) A class of new unitary invariant metrics: This paper generalized a class of new unitarily invariant metrics and give the upper-bounds and lower-bounds by (generalized) C-S decomposition. Based on that, two particular indentities of unitarily invariant metrics are obtained.3) Perturbation analysis of the canonical subspaces: This paper gives the least perturbation bound of the unitary factor on the unitary decomposition.Basing on that, we discuss the perturbation bounds of the canonical subspaces,which improve the results in [30].4) Two new proof: By applying (generalized) C-S decomposition,we prove two theorems of Sun's in [38],which improves his proof.