# Toeplitz矩陣

$\begin{bmatrix} a & b & c & d & k \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ j & h & g & f & a \end{bmatrix}$

$A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\ a_{1} & a_0 & a_{-1} & \ddots & & \vdots \\ a_{2} & a_{1} & \ddots & \ddots & \ddots& \vdots \\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix}$

$A_{i,j} = a_{i-j}.$

## 性質

$Ax=b$

$AU_n-U_mA,$

$AU_n-U_mA= \begin{bmatrix} a_{-1} & \cdots & a_{-n+1} & 0 \\ \vdots & & & -a_{-n+1} \\ \vdots & & & \vdots \\ 0 & \cdots & & -a_{n-n-1} \end{bmatrix}$

## 註

Toeplitz 系統 $Ax=b$ 可以用 Levinson-Durbin Algorithm 解，需時 Θ($n^2$) 。 呢套算法嘅變種，可證係喺 James Bunch 嘅意義上弱穏定(weakly stable) 嘅，(即係話，喺 well-condition 嘅線性系統，佢哋有 numerical stability )。

Toeplitz 矩陣同富理埃級數關係好密，因為「乘以一三角多項式」嘅 算子質埋入一有限維嘅空間時，可以用呢種矩陣表示。

If a Toeplitz matrix has the additional property that $a_i=a_{i+n}$, then it is a circulant matrix.

Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are centrosymmetric.

$\begin{matrix}y & = & x \ast h \\ & = & \begin{bmatrix}h_1\\h_2 \\h_3\\ \vdots \\ h_{m-1} \\h_m \\ \end{bmatrix} \begin{bmatrix}x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & 0& \ldots & 0 \\ 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & \ldots & 0 \\ 0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \ldots & 0 \\ 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} & 0 \\ 0 & 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} \\ \end{bmatrix} \end{matrix}$.

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc [1].

## 出面網頁

1. Using Toeplitz matrices in MATLAB [1]