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Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien, Hiroshi Nakazato (2016)

Czechoslovak Mathematical Journal

The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1 . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0 , 1 , and present an elementary computation...

Computing the numerical range of Krein space operators

Natalia Bebiano, J. da Providência, A. Nata, J.P. da Providência (2015)

Open Mathematics

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined...

Convergence of L p -norms of a matrix

Pavel Stavinoha (1985)

Aplikace matematiky

a recurrence relation for computing the L p -norms of an Hermitian matrix is derived and an expression giving approximately the number of eigenvalues which in absolute value are equal to the spectral radius is determined. Using the L p -norms for the approximation of the spectral radius of an Hermitian matrix an a priori and a posteriori bounds for the error are obtained. Some properties of the a posteriori bound are discussed.

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