Linear and multilinear algebra; matrix theory Definition

Toeplitz matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $m \times n$ matrix over $R$ . The matrix $A$ is a Toeplitz matrix when $a_{ij} = a_{i+1, j+1}$ for all $1 \le i < m$ and $1 \le j < n$ . Equivalently, there are elements $t_k \in R$ , for $1 - n \le k \le m - 1$ , with $a_{ij} = t_{i-j}$ for all $i, j$ .

Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 5 & 4 & 1 \end{pmatrix}$ is Toeplitz, with $t_0 = 1$ , $t_{-1} = 2$ , $t_{-2} = 3$ , $t_1 = 4$ , $t_2 = 5$ . Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not Toeplitz, because $a_{11} = 1 \ne 4 = a_{22}$ . For $m = n$ , the circulant matrices are the Toeplitz matrices with $t_k = t_{k-n}$ for $1 \le k \le n-1$ . If $J$ is the $m \times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Toeplitz exactly when $JA$ is a Hankel matrix.

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Matrix concept
Ring concept
Toeplitz matrix definition

Tag	`3G7WAP9DJJ`
URI	`m2://3G7WAP9DJJ`
Kind	definition
Writing	W0 stub
MSC	15B05
Defines	Toeplitz matrix
Uses	Ring, Matrix
Changed	2 November 2026

Toeplitz matrix

Definition

Remarks

Reading path

Further reading

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