Some Estimates of the Proper Values of Matrices
Miroslav Fiedler · Journal of the Society for Industrial and Applied Mathematics · 1965
Introduction. It is the purpose of this paper to obtain somewhat stronger results than those found previously by V. Ptak and the author [1]. We shall use here the same notation. Especially, we assume that we are given an n-dimensional (n > 1) complex linear space X whose vectors are denoted as row-vectors x = (xl , x2, , Xn). The vectors y = (x2, * -, YXn) form an (n 1)-dimensional space Y. Let g(y) be a norm in Y. If B is an operator in Y, its norm g(B) is (as usually) defined as sup; g(xB) for g(x) 1. The adjoint norm of a column (n1)-rowed vector z' is defined as g'(z') = supy I yz' I for g(y) < 1. We shall also write simply B k instead of B kI, where I is the identity operator. In the sequel, we shall use the following properties of the reciprocal norms of operators: