The Factorization of a Matrix as the Commutator of Two Matrices

Abstract

Let P = f " + (- I ,,) , the direct sum of the p x p identity matrix and the negative of the q x q ide n­ t ity matrix. The following th eo re m is proved. TH EOHEM: If X = cZ where Z is a 4 x 4 P-orthogonal , P-skew-symmetric matrix and Ie I .;;; 2, there exist matrices A an.d B, both of which are P-orthogollal and P-skew-symmetric, sach that X = AB - BA. Methods for o btaining certain matrices whi c h sati s fy X = AB - BA are given. Methods are a lso give n fo r de terminin g pairs of a nti co mmuting P -o rth"gona l, P -s kew-symmetri c matrices.