On the two-sided perplectic singular value decomposition and perplectically diagonalizable matrices

Abstract

Necessary and sufficient conditions for a square matrix [Formula: see text] to be written as [Formula: see text], where [Formula: see text] is a diagonal matrix, and [Formula: see text] and [Formula: see text] are isometries of either the Minkowski indefinite inner product or the Symplectic scalar product, are known. We give the analogue of the preceding decomposition to the perplectic scalar product [Formula: see text], where [Formula: see text] is the backward identity matrix. We also give a perplectic analogue of the Takagi factorization for symmetric matrices. As a related result, we give necessary and sufficient conditions for a perplectic matrix to be diagonalizable by a perplectic matrix, and we show that every perplectic matrix is a product of perplectic matrices which are diagonalizable by a perplectic matrix.