The ϕS polar decomposition

Abstract

We say that A has a φ S polar decomposition if A = QT for some φ S orthogonal Q ∈ M n (C) and some φ S symmetric T ∈ M n (C). Let V S = {A ∈ M n (C) : φ S (φ S (A)) = A}. We show that every nonsingular A ∈ V S has a φ S polar decomposition. We determine conditions for which an A ∈ M n (C) has a φ S polar decomposition. We also determine the possible Jordan Canonical Forms of a φ S orthogonal matrix and of a φ S skew symmetric matrix.