Toeplitz Matrix

• categories: Linear algebra, Matrix

Definition

A Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant. Formally, a Toeplitz matrix $T\in {\mathbb{R}}^{m\times n}$ satisfies:

$$T_{ij}=T_{i+1,j+1},\forall i,j\text{(elements depend only on }j-i\text{)}.$$

For a Toeplitz matrix, the entries can be defined using the first row and the first column.

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Example

General Form:

A Toeplitz matrix has the form:

$$T=\left[\begin{array}{ccccc}{t}_0& t_1& t_2& \cdots & t_{n-1}\\ t_{-1}& t_0& t_1& \cdots & t_{n-2}\\ t_{-2}& t_{-1}& t_0& \cdots & t_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ t_{-m+1}& t_{-m+2}& t_{-m+3}& \cdots & t_0\end{array}\right].$$

Specific Example:

For $m=n=4$, let the first row be $\{1,2,3,4\}$ and the first column be $\{1,5,6,7\}$:

$$T=\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 1& 2& 3\\ 6& 5& 1& 2\\ 7& 6& 5& 1\end{array}\right].$$

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Key Properties

1. Storage Efficiency:
   A Toeplitz matrix can be stored with $m+n-1$ elements, rather than $m\times n$, because it is fully defined by its first row and column.

2. Matrix-Vector Multiplication:
   Multiplying a Toeplitz matrix by a vector can be performed efficiently using the Fast Fourier Transform (FFT), in $O(n\log n)$ time for $n\times n$ matrices.

3. Circulant Relation:
   If the first row of a Toeplitz matrix is extended cyclically, it becomes a Circulant matrix, which is a special case of a Toeplitz matrix.

4. Structure in Linear Systems:
   Linear systems involving Toeplitz matrices, $T\mathbf{x}=\mathbf{b}$, can be solved efficiently using specialized algorithms like Levinson recursion in $O(n^2)$ time.

5. Symmetry:

   • If the first row and first column satisfy $t_{-k}=t_k$, the Toeplitz matrix is symmetric.

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Applications

1. Signal Processing:
   Toeplitz matrices arise in convolution operations and are used in autocorrelation and time-series analysis.

2. Numerical Linear Algebra:
   Efficient algorithms leverage Toeplitz structure to speed up operations like matrix-vector products and eigenvalue computations.

3. Control Theory:
   Appears in solving control and system identification problems.

4. PDEs and Integral Equations:
   Arise in discretization of problems with translation-invariant kernels.

Relation to Circulant Matrices

If the Toeplitz matrix is made periodic by assuming $t_{-k}=t_{n-k}$, it becomes a circulant matrix, which has diagonalization properties using the Discrete Fourier Transform (DFT). This property is exploited in solving convolution problems.