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    Item type:Publication, Access status: without files ,
    The determinant of a complex matrix and Gershgorin circles
    (Soc., 2019)
    Bünger, Florian
    ;
    Rump, Siegfried M.
    Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affrmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.
    Publicationtype: Journal Article
      188
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    Positive entries of stable matrices
    (2005-01-01)
    Friedland, Shmuel
    ;
    Hershkowitz, Daniel
    ;
    Rump, Siegfried M.
    The question of how many elements of a real positive stable matrix must be positive is investigated. It is shown that any real stable matrix of order greater than 1 has at least two positive entries. Furthermore, for every stable spectrum of cardinality greater than 1 there exists a real matrix with that spectrum with exactly two positive elements, where all other elements of the matrix can be chosen to be negative.
    Publicationtype: Journal Article
      39
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    Variational characterizations of the sign-real and the sign-complex spectral radius
    (ILAS, The International Linear Algebra Society, 2002)
    Rump, Siegfried M.
    The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical 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 the corresponding inf-max characterization as well as variational characterizations of the generalized (real and complex) spectral radius are presented. Again those are almost identical to the corresponding results in classical Perron-Frobenius theory.
    Publicationtype: Journal Article
      54