Palindromic Linearizations of Palindromic Matrix Polynomials of Odd Degree Obtained from Fiedler-Like Pencils
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Palindromic Linearizations of Palindromic Matrix Polynomials of Odd Degree Obtained from Fiedler-Like Pencils Ranjan Kumar Das1 · Rafikul Alam1 Received: 1 September 2019 / Accepted: 11 June 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract Palindromic matrix polynomials arise in many applications. Structure-preserving linearizations of palindromic matrix polynomials have been proposed in the literature so as to preserve the spectral symmetry in the eigenvalues. We present a new family of palindromic strong linearizations of a palindromic matrix polynomial of odd degree. A salient feature of the new family is that it allows the construction of banded palindromic linearizations of block-bandwidth k + 1 for any k = 0 : m − 2, where m is the degree of the palindromic matrix polynomial. Low bandwidth palindromic pencils may be useful for numerical computations. Our construction of the new family is based on Fiedler companion matrices associated with matrix polynomials and the construction is operation-free. Moreover, the new family of palindromic pencils allows operation-free recovery of eigenvectors and minimal bases, and an easy recovery of minimal indices of matrix polynomials from those of the palindromic linearizations. We also present an operation-free algorithm for construction of palindromic pencils belonging to the new family. Keywords Palindromic matrix polynomial · Fiedler pencil · Linearization · Eigenvalue · Eigenvector · Minimal basis · Minimal indices Mathematics Subject Classification (2010) 65F15 · 15A57 · 15A18 · 65F35
1 Introduction Structured polynomial eigenvalue problems arise in many applications. The structure (e.g., symmetric, skew-symmetric, palindromic, even, odd) of a matrix polynomial often induces Dedicated to Professor Volker Mehrmann on the occasion of his 65th birthday. Rafikul Alam
[email protected]; [email protected] Ranjan Kumar Das [email protected] 1
Department of Mathematics, IIT Guwahati, Guwahati,781039, India
R.K. Das, R. Alam
spectral symmetry in the eigenvalues. The spectral symmetry in the eigenvalues has physical significance and hence it is important that a linearization preserves the spectral symmetry [35]. Volker Mehrmann played a pivotal role in bringing into focus the importance of structure-preserving methods for solving structured eigenvalue problems, especially the need for the development of structure-preserving linearizations for structured polynomial eigenvalue problems and structure-preserving algorithms for solving structured linear eigenvalue problems, and structured eigenvalue perturbation theory, see [5, 7, 35–39] and the references therein. His pioneering research work has inspired many researchers to undertake research work in the areas of numerics of structured eigenvalue problems and structured eigenvalue perturbation theory; see, for example, [1–4, 8–10, 15, 16, 21, 23, 27, 28, 31]. Algorithms for generalized palindromic eigenvalue problems have been discussed in [29, 33]. The focus
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