Persymmetric matrix

In mathematics, persymmetric matrix may refer to:

1. a square matrix which is symmetric in the northeast-to-southwest 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

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

a_ij = a_{n − j + 1,n − i + 1} for all i, j.^[1]

For example, 5-by-5 persymmetric matrices are of the form

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

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 = (a_ij) is persymmetric if a_ij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form

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.

References