Abstract
This paper concerns the study of matrix polynomials of arbitrary degree. In terms of \(L(\lambda )=\lambda ^{r}I-\sum \limits _{j=1}^{r}\lambda ^{r-j}\mathcal {C}_{j}\) with or without commuting coefficients (\(\mathcal {C}_{i}\mathcal {C}_{j}=\mathcal {C}_{j}\mathcal {C}_{i}, \ \ \ or\ \ \ \mathcal {C}_{i}\mathcal {C}_{j}\ne \mathcal {C}_{j}\mathcal {C}_{i}\ \ \ for\ \ \mathcal {C}_i\in \mathbb {C}^{t\times t},\ \ i,j=1,\ldots , r\)) by determinant of block Toeplitz-Hessenberg matrices.
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References
Boege, T.: Gaussoids are two-antecedental approximations of Gaussian conditional independence structures. Ann. Math. Artif. Intell. 90, 645–673 (2022)
Boyd, J.P.: Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding. SIAM J. Numer. Anal. 40(5), 1666–1682 (2002)
Collao, M., Salas, M., Soto, R.L.: Toeplitz nonnegative realization of spectra via companion matrices. Special Matrices. 7, 230–245 (2019)
Datta, B.N.: Numerical Linear Algebra and Applications, 2nd edn. SIAM, Philadelphia (2010)
Gong, Z., Aldeen, M., Elsner, L.: A note on a generalized Cramer’s rule. Linear Algebra Appl. 340(13), 253–254 (2002)
Mackey, D.S., Mackey, N., Petrovic, S.: Is every matrix similar to a Toeplitz matrix? Linear Algebra Appl. 297, 87–105 (1999)
Merca, M.: A note on the determinant of a Toeplitz-Hessenberg matrix. Special Matrices. 1, 10–16 (2013)
Shams Solary, M.: Computational properties of pentadiagonal and anti-pentadiagonal block band matrices with perturbed corners. Soft Comput. 24, 301–309 (2020)
Tismenetsky, M.: Determinant of block Toeplitz band matrices. Linear Algebra Appl. 85, 165–184 (1987)
Trefethen, L.N.: Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics, (2018)
Wimmer, H.K.: Similarity of block companion and block Toeplitz matrices. Linear Algebra Appl. 343–344, 381–387 (2002)
Acknowledgements
Special thanks go to Ricardo L. Soto and Ira Gessel for enlightening conversations that inspired this work. Also, the author is grateful to the referees for constructive comments and suggestions that helped to improve the presentation.
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Solary, M.S. From matrix polynomial to determinant of block Toeplitz–Hessenberg matrix. Numer Algor 94, 1421–1434 (2023). https://doi.org/10.1007/s11075-023-01541-w
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DOI: https://doi.org/10.1007/s11075-023-01541-w