Polar Decomposition And Gauss Decomposition

Abstract

Abstract Every real positive definite symmetric matrix x can be diagonalized, that is, there exist an orthogonal matrix u and a positive diagonal matrix d such that x = udu′. This gives a decomposition of x which we call the polar decomposition because of its similarity to polar coordinates in the plane. Another decomposition of x is the Gauss decomposition x = tt′, where t is a lower triangular matrix with positive diagonal entries. In this chapter we describe the generalizations of these decompositions to arbitrary Euclidean Jordan algebras. The main results are the formulas of Theorem VI.2.3 and Theorem Vl.3.9, which express the integral of a function on Ω in terms of the polar decomposition and the Gauss decomposition.