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    Spectral properties of one class of sign-symmetric matrices
    (Maltepe Üniversitesi, 2009) Kushel, Olga Y.
    A matrix A of a linear operator A : R n ? R n is called J –sign-symmetric, if there exists such a subset J ? {1, . . . , n}, that the inequality aij ? 0 follows from the inclusions i ? J , j ? J c and j ? J , i ? J c for any two numbers i, j, and one of the inclusions i ? J , j ? J c or j ? J , i ? J c follows from the strict inequality aij < 0 (here J c = {1, . . . , n} \ J ). This definition is a generalization of the well-known definition of positive matrices, which are widely used in economics, mechanics, biology and other branches of science. Let A be a J –sign-symmetric matrix, and let J be a subset of {1, . . . , n} in the definition of J –sign-symmetricity. Let its second compound matrix A(2) also be a J –sign-symmetric matrix. Let Je be a subset of {1, . . . , C2 n} in the definition of J –sign-symmetricity for the matrix A(2). Let us construct the set W(J , Je) ? ({1, . . . , n} × {1, . . . , n}) by the following way: (i, j) ? W(J , Je) if and only if one of the following two cases takes place: (a) both the numbers i, j belong either to the set J , or to the set J c , besides, if i < j, then the number of the pair (i, j) in the lexicographic numeration belongs to the set Je, and if i > j, then the number of the pair (j, i) belongs to the set Jec = {1, . . . , C2 n} \ Je; (b) one of the numbers i, j belongs to the set J , the other belongs to the set J c , besides, if i < j, then the number of the pair (i, j) belongs to the set Jec , and if i > j, then the number of the pair (j, i) belongs to the set Je. Such a set is not uniquely defined, but there is a finite number of different ways of its constructing. The set W(J , Je) is called transitive if the inclusion (i, k) ? W(J , Je) follows from the inclusions (i, j) ? W(J , Je) and (j, k) ? W(J , Je) for any indices i, j, k ? {1, . . . , n}. Theorem 0.1 Let the matrix A of a non-zero linear operator A be J –sign-symmetric together with its second compound matrix A(2). Then the operator A has a positive eigenvalue ?1 = ?(A). More than that, if ?1 is a simple eigenvalue, then one of the following two cases takes place: (1) If at least one of the possible sets W(J , Je) is transitive, then the second in modulus eigenvalue ?2 of the operator A is nonnegative and different in modulus from the first eigenvalue ?1. (2) If all the possible sets W(J , Je) are not transitive, there there is an odd number k of eigenvalues on the spectral circle |?| = ?(A). All of them are simple and coincide with kth roots of (?(A))k .