Persymmetric matrix

In mathematics, persymmetric matrix may refer to: 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.
 * 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.

Definition 1
Let $A = (a_{ij})$ be an $n&thinsp;×&thinsp;n$ matrix. The first definition of persymmetric requires that $$a_{ij} = a_{n-j+1,\,n-i+1}$$ for all $i, j$. For example, 5&thinsp;×&thinsp;5 persymmetric matrices are of the form $$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\ a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\ a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\ a_{51} & a_{41} & a_{31} & a_{21} & a_{11} \end{bmatrix}.$$

This can be equivalently expressed as $AJ = JA^{T}$ where $J$ is the exchange matrix.

A third way to express this is seen by post-multiplying $AJ = JA^{T}$ with $J$ on both sides, showing that  $A^{T}$ rotated 180 degrees  is identical to $A$: $$A = J A^\mathsf{T} J.$$

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. 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 $$A = \begin{bmatrix} r_1 & r_2 & r_3 & \cdots & r_n \\ r_2 & r_3 & r_4 & \cdots & r_{n+1} \\ r_3 & r_4 & r_5 & \cdots & r_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1} \end{bmatrix}.$$ A persymmetric determinant is the determinant of a persymmetric matrix.

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