A Generalization of the Polar Representation of Nonsingular Matrices

where P is a positive definite hermitian matrix and U is a unitary matrix. In this polar representation of the matrix A, as it is called, the two matrices P and U are unique. Since the matrix P is positive definite and nonsingular, it has the same signature as the identity matrix E while the unitary matrix U is a conjunctive automorph of E. From (1) we may deduce a somewhat similar representation of A in terms of a positive definite hermitian matrix and a conjunctive automorph, not of E, but of any nonsingular positive definite hermitian matrix. Let i l b e a nonsingular hermitian matrix which is positive definite, so that there exists a nonsingular matrix Q satisfying