# Toeplitz矩陣

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

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

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

## 性質

${\displaystyle Ax=b}$

${\displaystyle AU_{n}-U_{m}A,}$

${\displaystyle AU_{n}-U_{m}A={\begin{bmatrix}a_{-1}&\cdots &a_{-n+1}&0\\\vdots &&&-a_{-n+1}\\\vdots &&&\vdots \\0&\cdots &&-a_{n-n-1}\end{bmatrix}}}$

## 註

Toeplitz 系統 ${\displaystyle Ax=b}$ 可以用 Levinson-Durbin Algorithm 解，需時 Θ(${\displaystyle n^{2}}$) 。 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 ${\displaystyle a_{i}=a_{i+n}}$, then it is a circulant matrix.

Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.

${\displaystyle {\begin{matrix}y&=&x\ast h\\&=&{\begin{bmatrix}h_{1}\\h_{2}\\h_{3}\\\vdots \\h_{m-1}\\h_{m}\\\end{bmatrix}}{\begin{bmatrix}x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&0&0&\ldots &0\\0&x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&0&\ldots &0\\0&0&x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\ldots &\vdots &\vdots &\ldots &0\\0&\ldots &0&0&x_{1}&\ldots &x_{n-2}&x_{n-1}&x_{n}&0\\0&0&\ldots &0&0&x_{1}&\ldots &x_{n-2}&x_{n-1}&x_{n}\\\end{bmatrix}}\end{matrix}}}$.

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] 互聯網檔案館歸檔，歸檔日期2011年7月8號，.