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Self-inversive matrix polynomials with semisimple spectrum on the unit circle. (English) Zbl 1235.15008

Let \(A_i\in\mathbb C^{n\times n}\), \(P(z)=\sum_{i=1}^kA_iz^i\) and \(\det A(z)\) is not identically zero. \(\lambda\in\mathbb C\) is said to be a characteristic value of \(P(z)\) if \(\det P(\lambda)=0\), \(\lambda\) is said to be semisimple if the corresponding elementary divisors are linear. In the paper a condition under which all characteristic values of a matrix polynomial are semisimple and lie on the unit circle is given.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
15A54 Matrices over function rings in one or more variables
39A06 Linear difference equations
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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