An improvement of Ky Fan theorem for matrix eigenvalues

Some improvements of a Ky Fan theorem are presented for matrix eigenvalue inclusion problems. Subsequently, we investigate the relationships between K-0 (A), the union of Ky Fan theorem sets, S(A), the union of the Theorem 2.1 sets, and K-1(A), the union of the Theorem 2.2 sets, for eigenvalue inclusions of an n x n complex matrix A. If a(A) denotes the spectrum of A, we show that sigma (A) subset of S (A) subset of K-1 (A) subset of K-0 (A) is valid for any weakly irreducible n x n complex matrix A with n >= 2. (c) 2005 Elsevier Ltd. All rights reserved.