What are practical examples of Toeplitz matrices?

Question

A Toeplitz matrix is one in which each descending diagonal from left to right is constant. Given that structure, matrix operations are sometimes much faster. Where are Toeplitz matrices likely to occur?

Answer 1

One application is linear difference equations with constant coefficients. For example, suppose that we have the (minimalistic) moving average (MA) filter $y_k = \frac12 (x_k + x_{k-1})$ with $x_{-1} = 0$. In matrix form,

$$ \begin{bmatrix} y_0 \\ y_1 \\ y_2 \end{bmatrix} = \frac12 \begin{bmatrix} 1 & & \\ 1 & 1 & \\ & 1 & 1 \end{bmatrix} \begin{bmatrix} x_0 \\ x_1 \\ x_2 \end{bmatrix} $$

where the matrix is

• lower triangular because of causality

• bidiagonal because this MA filter is of length $2$

• Toeplitz because the difference equation has constant coefficients.