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Quadratic realizability of palindromic matrix polynomials

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2019-04-15
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Elsevier
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Let L = (L-1 , L-2) be a list consisting of a sublist L(1 )of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L-2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Mobius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.
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Matrix polynomials, Quadratic realizability, Elementary divisors, Minimal indices, Quasi-canonical form, Quadratifications, T-palindromic, Inverse problem
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De Terán, F., Dopico, F. M., Mackey, D. S. & Perović, V. (2019). Quadratic realizability of palindromic matrix polynomials. Linear Algebra and Its Applications, 567, pp. 202–262.