# User guide to the CQT toolbox

## Contents

## Introduction

The CQT toolbox implements the arithmetic of $\mathcal{QT}$ matrices, which are defined as the set of matrices obtained summing a semi-infinite Toeplitz matrix and a correction to the top-left corner.

A $\mathcal{QT}$ matrix has the structure $A = T(a(z))+ E_a$ where

$$ T(a(z)) := \left[ \begin{array}{cccc} a_0 & a_1 & a_2 & \cdots \\ a_{-1} & a_0 & \ddots & \ddots \\ a_{-2} & \ddots & \ddots\\ \vdots & \ddots \end{array} \right], $$

and $E_a$ is a compact correction. Although $A$ is a matrix of infinite size, we assume that its symbol $a(z)$ admits a convergent Laurent series and that ${\rm vec}(E)$ is in $l^1(N)$. This allows to operate numerically on these objects by appropriately truncating these matrices.

The CQT toolbox allows to represent and operate these matrices directly in the MATLAB environment.

If you are unfamiliar with QT matrices, visit the following link to understand how to interact with infinite matrices in MATLAB.

## Markov chains with infinite-dimensional state spaces

$\mathcal{QT}$ matrices are a natural tool for analyzing Markov chains and random walks on infinite state spaces. We present a few examples of such analyses here.

- Computing invariant probabilities for certain Quasi-Birth-and-Death Markov processes involving quadratic matrix equations.
- Solving a quadratix matrix equation using cyclic reduction and ${\mathcal QT} matrices.

## Matrix functions of infinite matrices

- Computing the matrix square root of semi-infinite Toeplitz matrices with a low-rank correction.

## Computing eigenvalues of QT matrices

- Computing eigenvalues of ${\mathcal QT}$ matrices.