Toeplitz Matrix
Table of Contents
1 toeplitz matrix
For such a long time, I have it on my list to write something for Toeplitz matrix. I want to derive the fundamental theorems on the asymptotic behavvior of eigenvalues, inverses, and products of banded Toeplitz matrices. To achieve this goal, I read the famous paper Toeplitz and Circulant Matrices: A review by Robert M. Gray Gray_2005 . I am fond of matrices with the style like Toeplitz matrices imagining that there must be something special existing in the square matrices. During writing this post, Matlab is also used to finish some mathematical test.
1.1 Toeplitz Matrices
A Toeplitz matrix \(T\) is an square matrix with size \(n\times n\). \(T_{n} = [t_{k,j}; k,j= 0,1,\ldots,n-1]\) where \(t_{k,j} = t_{k-j}\).
In matrix form, A toeplitz matrix can be expressed as:
\begin{equation} \label{eq:1} T_{n} = \begin{pmatrix} t_{0} & t_{-1} & t_{-2} &\ldots & t_{-(n-1)} \\ t_{1} & t_{0} & t_{-1} & \ldots &\vdots \\ t_{2} & t_{1} & t_{0} & \ldots &\vdots \\ \vdots & \vdots &\vdots & \ddots & \vdots \\ t_{n-1} & \ldots & \ldots & \ldots & t_{0} \end{pmatrix} \end{equation}Because we dive into the details of Toeplitz matrix, some keypoints of matrices are recaptured, such as the eigenvalues, Norms and asymptotically equivalent matrices.
1.2 Some Asymptotic Properties of Matrices
Most often, we can study the asymptotic behavior of complicated matrices by studying a more structured and simplier asymptotically equivalent matirx. This method is used in many fields. A toy model is studied comprehensively before a more complicated one comes in.
1.2.1 Eigenvalues
Scalor \(\alpha\) and vector \(x\) satisfied \(Ax = \alpha x\) are called the eigenvalue and the corresponding eigenvector. We can get the eigenvalues by solving
\begin{equation} \label{eq:2} \det(A-\alpha I) = 0 \end{equation}For the sake of simplicity, we assume that the eigenvalues are ordered in noincreasing format, i.e. \(\alpha_{1} > \alpha_{2} > \ldots \).
Any complex matrix \(A\) can be written as:
\begin{equation} \label{eq:3} A = URU^{*} \end{equation}where the asterisk \(*\) denotes conjugate transpose, \(U\) is unitary, i.e., \(U^{-1} = U^{*}\) and \(R\) is an upper triangular matrix. The eigenvalues of \(A\) are the principle diagonal elements of \(R\). if \(A\) is normal, i.e. if\(A^{*}A = AA^{*}\), then \(R\) is a diagonal matrix, which we denote as \(R = \diag(\alpha_{k})\). If \(A\) is Hermitian, i.e. if \(A^{*} = A\), then the eigenvalues are real. If a matrix is Hermitian, then it is also normal.
For the case of Hermitian matrices, a useful description of the eigenvalues is the variational description given by the Courant-Fischer theorem.
Define the Rayleigh quotient of an Hermitian matrix \(X\) and a vector (complex \(n\)-tuple ) x by
\begin{equation} \label{eq:4} R_{H}(x) = (x^{*}Hx)/ (x^{*}x) \end{equation}Let \(\eta_{M}\) and \(\eta_{m}\) be the maximum and minimum eigenvalues of \(H\), respectively.
Then
\begin{eqnarray} \label{eq:5} \eta_{m}&=& \min_{x} R_{H}(x) = \min_{x:x^{*}x = 1} x^{*}Hx \\ \eta_{M}&=& \max_{x} R_{H}(x) = \max_{x:x^{* }x = 1} x^{*}Hx \end{eqnarray}