5,779 research outputs found

    The Collatz-Wielandt quotient for pairs of nonnegative operators

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    In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A,B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator then one version of this quotient is the spectral radius of A. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient.Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-794

    Nonnegative definite hermitian matrices with increasing principal minors

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    A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. For m>1m>1 we show AA has increasing principal minors if and only if Aβˆ’1A^{-1} exists and its diagonal entries are less or equal to 1.Comment: 3 page
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