Instability indices for matrix polynomials |
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Authors: | Todd Kapitula Elizabeth Hibma Hwa-Pyeong Kim Jonathan Timkovich |
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Institution: | 1. Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI 49546, United States;2. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, United States |
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Abstract: | There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ?-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ?-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients. |
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Keywords: | 15A18 15A22 15B57 |
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