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  • Pure states, positive matrix polynomials and sums of hermitian squares
    Klep, Igor, matematik ; Schweighofer, Markus
    Let ▫$M$▫ be an archimedean quadratic module of real ▫$t \times t$▫ matrix polynomials in ▫$n$▫ variables, and let ▫$S \subseteq {\mathbb R}^n$▫ be the set of all points where each element of ▫$M$▫ ... is positive semidefinite. Our key finding is a natural bijection between the set of pure states of ▫$M$▫ and ▫$S \times {\mathbb P}^{t-1}({\mathbb R})$▫. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on ▫$S$▫, then it belongs to ▫$M$▫. We also discuss what happens for nonsymmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
    Source: Indiana University mathematics journal. - ISSN 0022-2518 (Vol. 59, no. 3, 2010, str. 857-874)
    Type of material - article, component part
    Publish date - 2010
    Language - english
    COBISS.SI-ID - 15816793

    Link(s):

    http://www.iumj.indiana.edu/oai/2010/59/4107/4107.xml

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