Palindromic matrix polynomials, matrix functions and integral representations
نویسندگان
چکیده
منابع مشابه
Smith Forms of Palindromic Matrix Polynomials
Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular T -palindromic matrix polynomials over arbitrary fields. The invariant polynomials are ...
متن کاملBackward errors and linearizations for palindromic matrix polynomials
We derive computable expressions of structured backward errors of approximate eigenelements of ∗-palindromic and ∗-anti-palindromic matrix polynomials. We also characterize minimal structured perturbations such that approximate eigenelements are exact eigenelements of the perturbed polynomials. We detect structure preserving linearizations which have almost no adverse effect on the structured b...
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The standard way to solve polynomial eigenvalue problems P (λ)x = 0 is to convert the matrix polynomial P (λ) into a matrix pencil that preserves its spectral information– a process known as linearization. When P (λ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P (λ) have certain symmetries that can be lost when using the classical first and second Frobenius comp...
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In this paper, we propose a palindromic quadratization approach, transforming a palindromic matrix polynomial of even degree to a palindromic quadratic pencil. Based on the (S + S−1)-transform and Patel’s algorithm, the structurepreserving algorithm can then be applied to solve the corresponding palindromic quadratic eigenvalue problem. Numerical experiments show that the relative residuals for...
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In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ”symmetric” polyno...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2011
ISSN: 0024-3795
DOI: 10.1016/j.laa.2010.09.013