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 К0(A), the union of Ky Fan theorem sets, S(A), the union of the Theorem 2.1 sets, and К1(A), the union of the Theorem 2.2 sets, for eigenvalue inclusions of an n × n complex matrix A. If σ(A) denotes the spectrum of A, we show that is valid for any weakly irreducible n × n complex matrix A with n ≥ 2.