Persymmetric matrix

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 = (a_ij) 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. For example, 5 × 5 persymmetric matrices are of the form $$A=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{14}\\ a_{31}& a_{32}& a_{33}& a_{23}& a_{13}\\ a_{41}& a_{42}& a_{32}& a_{22}& a_{12}\\ a_{51}& a_{41}& a_{31}& a_{21}& a_{11}\end{array}\right].$$

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: $$A=J{A}^{𝖳}J.$$

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

Further information: Hankel matrix

The second definition is due to Thomas Muir. 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 $$A=\left[\begin{array}{ccccc}{r}_1& r_2& r_3& \cdots & r_n\\ r_2& r_3& r_4& \cdots & r_{n+1}\\ r_3& r_4& r_5& \cdots & r_{n+2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ r_n& r_{n+1}& r_{n+2}& \cdots & r_{2n-1}\end{array}\right].$$ 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.

See also

• Centrosymmetric matrix