Core Concepts

A new matrix algebra approach is proposed to represent and efficiently compute with quasi-Toeplitz matrices, providing substantial computational advantages over existing methods.

Abstract

The paper analyzes the structural and computational properties of the linear space Pα spanned by the powers of a semi-infinite tridiagonal matrix Aα. It is shown that any matrix in Pα can be represented as the sum of a symmetric Toeplitz matrix and a Hankel matrix, where the Hankel component depends on the parameter α.
Key highlights:
A basis {P0, P1, P2, ...} of Pα is determined, where each Pn is the sum of a symmetric Toeplitz matrix and a Hankel matrix.
It is proven that Pα is a matrix algebra, and conditions are provided for Pα to be a Banach algebra.
For quasi-Toeplitz (QT) matrices with symmetric Toeplitz part, representing them as Pα(a) + KA, where Pα(a) is in the Pα algebra and KA is compact, is shown to be more efficient than the standard representation A = T(a) + EA.
Experimental results demonstrate substantial acceleration in solving matrix equations and computing matrix square roots for QT matrices using the new representation compared to existing methods.

Stats

The 2-norm and infinity norm of matrices in Pα are bounded if |α| ≤ 1, or if the generating Laurent series is analytic in an annulus around the unit circle.
The computational cost of matrix arithmetic operations in Pα is O((m+n)log(m+n)), where m and n are the degrees of the involved Laurent polynomials.

Quotes

"Representing a QT matrix A = T(a) + EA, where T(a) is symmetric, in the form Pα(a) + KA, for KA compact, is more convenient from the theoretical and the computational point of view."
"From the computational point of view, we compared the representation of QT matrices given in the CQT-Toolbox, designed in [6], with the new representation based on the algebra Pα. As test problems, we considered the solution of a matrix equation with QT coefficients performed by means of fixed point iterations, and the computation of the square root of a real symmetric QT matrix. In both cases, with the new representation we obtain substantial acceleration in the CPU time with respect to the standard representation given in [6]."

Key Insights Distilled From

by Dario Bini,B... at **arxiv.org** 05-07-2024

Deeper Inquiries

The proposed matrix algebra approach can be extended to handle non-symmetric quasi-Toeplitz matrices by considering a more general form of the matrices involved. In the context of quasi-Toeplitz matrices, the algebra Pα is defined based on the symmetric structure of the matrices. To handle non-symmetric matrices, we can modify the definition of the algebra to accommodate the asymmetry in the Toeplitz and Hankel components. This extension would involve representing the non-symmetric quasi-Toeplitz matrices as a sum of a non-symmetric Toeplitz matrix and a non-symmetric Hankel matrix, similar to the approach used for symmetric matrices. By adapting the basis and operations in the algebra Pα to account for non-symmetric structures, we can effectively handle non-symmetric quasi-Toeplitz matrices within this framework.

The Pα algebra, originally developed for quasi-Toeplitz matrices, has potential applications beyond this specific context in computational mathematics. One possible application is in signal processing, where structured matrices play a crucial role in various algorithms such as signal reconstruction, filtering, and compression. The Pα algebra could be leveraged to efficiently manipulate and analyze structured matrices arising in signal processing tasks. Additionally, in the field of optimization, structured matrices often appear in optimization problems, and the Pα algebra could offer a systematic way to handle these structured matrices, leading to improved computational efficiency and algorithm performance. Moreover, in the realm of machine learning and data analysis, structured matrices are prevalent in tasks like dimensionality reduction and feature extraction. The Pα algebra could provide a framework for handling structured matrices in these applications, enabling more efficient computations and enhanced algorithmic capabilities.

The insights from this work on the Pα algebra for quasi-Toeplitz matrices can inspire the development of novel matrix representations and computational techniques for other classes of structured matrices encountered in scientific computing and engineering. By studying the properties and operations within the Pα algebra, researchers can derive new approaches for handling structured matrices with specific patterns and characteristics. This knowledge can be applied to design efficient algorithms for matrix manipulation, inversion, and decomposition tailored to different classes of structured matrices. The concept of representing matrices as a sum of structured components, as seen in the Pα algebra, can be extended to other structured matrix classes, leading to the development of innovative techniques for solving matrix equations, optimizing matrix operations, and enhancing computational methods in various scientific and engineering disciplines.

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