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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2010, Volume 50, Number 5, Pages 805–816
(Mi zvmmf4871)
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On commutative algebras of Toeplitz-plus-Hankel matrices
Kh. D. Ikramov, Yu. O. Vorontsov Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992 Russia
Abstract:
It is known that the entire class of Hermitian Toeplitz matrices can be mapped into a subset of real Toeplitz-plus-Hankel matrices ($(T + H)$-matrices) by one and the same unitary similarity transformation. This fact is refined by showing that the resulting $(T + H)$-matrices are symmetric. Moreover, the symmetry is preserved if this similarity transformation is applied to arbitrary (rather than only Hermitian) Toeplitz matrices and even if it is applied to a much broader class of persymmetric matrices. Let the same similarity transformation be applied to the class of normal Toeplitz matrices. By examining the range of this transformation, commutative algebras are selected that consist of (complex) symmetric $(T + H)$-matrices; in addition, all the matrices in these algebras are normal. An algorithm is proposed for multiplying matrices belonging to these algebras. Its complexity is equivalent to that of multiplying two circulants of order $n$, which is several times less than the complexity of multiplying two general $(T + H)$-matrices.
Key words:
Toeplitz matrices, circulants, Hankel matrices, persymmetric matrices, Toeplitz-plus-Hankel matrices, Fast Fourier Transform.
Received: 18.09.2009 Revised: 01.12.2009
Citation:
Kh. D. Ikramov, Yu. O. Vorontsov, “On commutative algebras of Toeplitz-plus-Hankel matrices”, Zh. Vychisl. Mat. Mat. Fiz., 50:5 (2010), 805–816; Comput. Math. Math. Phys., 50:5 (2010), 766–777
Linking options:
https://www.mathnet.ru/eng/zvmmf4871 https://www.mathnet.ru/eng/zvmmf/v50/i5/p805
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