The ψS polar decomposition when the cosquare of S is normal

• Main Authors: Granario, Daryl Q., Merino, Dennis I., Paras, Agnes T.
• Format: text
• Published: Animo Repository 2016
• Subjects: Symmetric matrices; Decomposition (Mathematics); Orthogonal decompositions; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/11361
• Institution: De La Salle University

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.