On certain matrix algebras related to quasi-Toeplitz matrices
Dario Bini, Beatrice Meini
TL;DR
The paper introduces an $\alpha$-parametrized family of matrix algebras $\mathcal{P}_\alpha$ generated by powers of the semi-infinite matrix $A_\alpha$, showing elements can be written as $P_\alpha(a)=T(a)+H_\alpha(a)$ and that $\mathcal{P}_\alpha$ forms a Banach algebra. It then advocates representing symmetric QT matrices as $A=P_\alpha(a)+K_A$ rather than $T(a)+E$, enabling more efficient matrix arithmetic and function evaluation via $f(A)=P_\alpha(f(a))+K_f$, with reduced growth of the compact correction. The authors establish structural results, special cases (notably $\alpha\in\{-1,0,1\}$ and finite $m$), and discuss boundedness, analyticity requirements, and computational schemes (FFT-based, SWM) to support robust implementations. Numerical experiments show the new representation yields significant speedups over the CQT-toolbox while maintaining comparable numerical accuracy, highlighting practical impact for quadratic matrix equations and matrix square roots. Overall, the work provides a principled, algebraic reformulation that enhances both the theory and computation of symmetric QT matrices.
Abstract
Let $A_α$ be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, $(A_α)_{11}=α$, where $α\in\mathbb C$, and zero elsewhere. A basis $\{P_0,P_1,P_2,\ldots\}$ of the linear space $\mathcal P_α$ spanned by the powers of $A_α$ is determined, where $P_0=I$, $P_n=T_n+H_n$, $T_n$ is the symmetric Toeplitz matrix having ones in the $n$th super- and sub-diagonal, zeros elsewhere, and $H_n$ is the Hankel matrix with first row $[θα^{n-2}, θα^{n-3}, \ldots, θ, α, 0, \ldots]$, where $θ=α^2-1$. The set $\mathcal P_α$ is an algebra, and for $α\in\{-1,0,1\}$, $H_n$ has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices $\mathcal {QT}_S$, where, instead of representing a generic matrix $A\in\mathcal{QT}_S$ as $A=T+K$, where $T$ is Toeplitz and $K$ is compact, it is represented as $A=P+H$, where $P\in\mathcal P_α$ and $H$ is compact. It is shown experimentally that the matrix arithmetic obtained this way is much more effective than that implemented in the CQT-Toolbox of Numer.~Algo. 81(2):741--769, 2019.