We denote by F the field R of real numbers, the field C of complex numbers, or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be Hermitian if A = A* and unitary if AA* = In, where In is the n × n identity matrix. An n × n Hermitian matrix A is said to be positive definite (postive semi-definite resp.) if uAu* > 0(uAu* ≥ 0 resp.) for all u (╪ 0) in Fn. Here and in what follows we regard u as a 1 × n matrix and identify a 1 × 1 matrix with its single element. In the following we shall always use A and B to denote two n×n Hermitian matrices with elements in F, and we say that A and B can be diagonalized simultaneously if there exists an n×n non-singular matrix V with elements in F such that VAV* and VBV* are diagonal matrices. We shall use diag {A1, A2} to denote a diagonal block matrix with the square matrices A1 and A2 lying on its diagonal.