VARIATIONAL CHARACTERIZATIONS OF THE SIGN-REAL AND THE SIGN-COMPLEX SPECTRAL RADIUS

Abstract. The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the Perron-Frobenius theory for nonnegative matrices to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one correspondences to classical Perron-Frobenius theory. In this paper we prove variational characterizations of the generalized (real and complex) spectral radius which are again almost identical to the corresponding one in classical Perron-Frobenius theory. 1. Introduction. Denote IR+: = {x ≥ 0: x ∈ IR}, and let IK ∈ {IR+, IR,C}. The generalized spectral radius is defined by ρIK(A): = max{|λ | : ∃ 0 6 = x ∈ IKn, ∃λ ∈ IK, |Ax | = |λx|} for A ∈Mn(IK).(1) Note that absolute value and comparison of matrices and vectors are always to be understood componentwise. For example, A ≤ |C | for A ∈Mn(IR), C ∈Mn(C) is equivalent to Aij ≤ |Cij | for all i, j