Linear and multilinear algebra; matrix theory Definition

Hankel 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 Hankel matrix when $a_{i, j+1} = a_{i+1, j}$ for all $1 \le i < m$ and $1 \le j < n$ . Equivalently, there are elements $h_k \in R$ , for $2 \le k \le m + n$ , with $a_{ij} = h_{i+j}$ for all $i, j$ .

Remarks

Example: $\begin{pmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 5 \end{pmatrix}$ is Hankel, with $h_2, \dots, h_6 = 1, 1, 2, 3, 5$ ; each anti-diagonal is constant. Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not Hankel, because $a_{12} = 2 \ne 3 = a_{21}$ . A square Hankel matrix is symmetric, since $a_{ij} = h_{i+j} = a_{ji}$ . The Hilbert matrix, with $h_k = 1/(k-1)$ , is a Hankel matrix. If $J$ is the $m \times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Hankel exactly when $JA$ is a Toeplitz matrix.

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Matrix concept
Ring concept
Hankel matrix definition

Tag	`TRRREQX56B`
URI	`m2://TRRREQX56B`
Kind	definition
Writing	W0 stub
MSC	15B05
Defines	Hankel matrix
Uses	Ring, Matrix
Changed	2 November 2026

Hankel matrix

Definition

Remarks

Reading path

Further reading

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