On the exponential of semi-infinite quasi-Toeplitz matrices

TL;DR

This paper investigates the exponential of semi-infinite quasi-Toeplitz matrices, proving it retains a quasi-Toeplitz structure and providing an efficient computational algorithm, with potential extensions to other functions and finite matrices.

Contribution

It establishes that the exponential of a semi-infinite quasi-Toeplitz matrix is also quasi-Toeplitz and introduces an effective method for its computation.

Findings

Exponential of semi-infinite quasi-Toeplitz matrices remains quasi-Toeplitz.

An efficient algorithm for computing the exponential is developed.

Results extend to other functions and finite matrices.

Abstract

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued function defined for $|z|=1$, such that $\sum_{i\in\mathbb Z}|ia_i|<\infty$, and let $E=(e_{i,j})_{i,j\in\mathbb {Z}^+}$ be such that $\sum_{i,j\in\mathbb{Z}^+}|e_{i,j}|<\infty$. A semi-infinite quasi-Toeplitz matrix is a matrix of the kind $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)$, that is, $t_{i,j}=a_{j-i}$ for $i,j\in\mathbb Z^+$. We analyze theoretical and computational properties of the exponential of $A$. More specifically, it is shown that $\exp(A)=T(\exp(a))+F$ where $F=(f_{i,j})_{i,j\in\mathbb{Z}^+}$ is such that $\sum_{i,j\in\mathbb{Z}^+}|f_{i,j}|$ is finite, i.e., $\exp(A)$ is a semi-infinite quasi-Toeplitz matrix as well, and an effective algorithm for its computation is given. These results can be extended from the function $\exp(z)$ to any function $f(z)$ satisfying mild conditions, and can be applied to finite quasi-Toeplitz matrices.