# Persymmetric matrix

Short description: Square matrix symmetric about its anti-diagonal

In mathematics, persymmetric matrix may refer to:

1. a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or
2. a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.

The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition.

## Definition 1

Symmetry pattern of a persymmetric 5 × 5 matrix

Let A = (aij) be an n × n matrix. The first definition of persymmetric requires that

$\displaystyle{ a_{ij} = a_{n-j+1,\,n-i+1} }$ for all i, j.[1]

For example, 5 × 5 persymmetric matrices are of the form

$\displaystyle{ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\ a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\ a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\ a_{51} & a_{41} & a_{31} & a_{21} & a_{11} \end{bmatrix}. }$

This can be equivalently expressed as AJ = JAT where J is the exchange matrix.

A third way to express this is seen by post-multiplying AJ = JAT with J on both sides, showing that AT rotated 180 degrees is identical to A:

$\displaystyle{ A = J A^\mathsf{T} J. }$

A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called bisymmetric matrices.

## Definition 2

The second definition is due to Thomas Muir.[2] It says that the square matrix A = (aij) is persymmetric if aij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form

$\displaystyle{ A = \begin{bmatrix} r_1 & r_2 & r_3 & \cdots & r_n \\ r_2 & r_3 & r_4 & \cdots & r_{n+1} \\ r_3 & r_4 & r_5 & \cdots & r_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1} \end{bmatrix}. }$

A persymmetric determinant is the determinant of a persymmetric matrix.[2]

A matrix for which the values on each line parallel to the main diagonal are constant is called a Toeplitz matrix.