The ψS polar decomposition when the cosquare of S is normal

Added Title

The psi S polar decomposition when the cosquare of S is normal

College

College of Science

Department/Unit

Mathematics and Statistics Department

Document Type

Article

Source Title

Linear Algebra and its Applications

Volume

495

First Page

51

Last Page

66

Publication Date

2016

Abstract

Let a nonsingular S ∈ Mn (C) be given. For a nonsingular A ∈ Mn (C), set ψS (A) = S−1A−1S. We say that an A is ψS orthogonal if ψS (A) = A−1 and we say that A is ψS symmetric if ψS (A) = A. For a possibly singular B ∈ Mn (C), we say that B is ψS orthogonal if S−1BS = B; we say that B has a ψS polar decomposition if B = RE for some (possibly singular) ψS orthogonal R and (necessarily nonsingular) ψS symmetric E. If S = I, then the ψS polar decomposition is the real-coninvolutory decomposition. We show that if A is nonsingular, then A has a ψS polar decomposition if and only if A commutes with SS. Because S is nonsingular, the cosquare of S (that is, S−T S) is normal if and only if SS is normal [11, Theorem 5.2]. In this case, we show that a possibly singular A ∈ Mn (C) has a ψS polar decomposition if and only if (a) rank (A) and rank SS − λI A have the same parity for every negative eigenvalue λ of SS, and (b) the ranges of SA and A are the same.

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Disciplines

Mathematics

Keywords

Symmetric matrices; Decomposition (Mathematics); Orthogonal decompositions

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