Maximum Principles in matrix theory

Abstract

Unless the contrary is stated, all matrices are understood to be complex and of type n × n. The transposed conjugate of A is denoted by A*. The non-negative square roots of the characteristic roots of A*A are called the singular values of A; they will be denoted by st(A), i = 1, …, n, where s1(A)≥…≥ sn(A). The symbol [A]k denotes the k × k submatrix standing in the upper left-hand corner of A. We shall write Ei(z1, …, zn) for the j-th elementary symmetric function of z1..., zn, and E1(A) for the j-th elementary symmetric function of the characteristic roots of A. It is understood that, throughout, 1≥j≥k≥n.