On certain matrix algebras related to quasi-Toeplitz matrices
Dario Bini, Beatrice Meini

TL;DR
This paper characterizes a specific algebra of matrices related to quasi-Toeplitz matrices, providing a new representation that improves computational efficiency in matrix arithmetic.
Contribution
It introduces a novel algebraic framework for symmetric quasi-Toeplitz matrices, enabling more effective matrix computations compared to previous methods.
Findings
The algebra $\\mathcal{P}_\alpha$ is explicitly constructed and shown to be closed under multiplication.
A new representation of symmetric quasi-Toeplitz matrices as $A=P+H$ improves computational efficiency.
Experimental results demonstrate superior performance of the new method over existing tools.
Abstract
Let be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, , where , and zero elsewhere. A basis of the linear space spanned by the powers of is determined, where , , is the symmetric Toeplitz matrix having ones in the th super- and sub-diagonal, zeros elsewhere, and is the Hankel matrix with first row , where . The set is an algebra, and for , has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices , where, instead of representing a generic matrix as…
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Taxonomy
TopicsAdvanced Topics in Algebra · Matrix Theory and Algorithms · Advanced Algebra and Logic
