Persymmetric matrix explained

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

Let be an matrix. The first definition of persymmetric requires that $$a\_\; =\; a\_$$ for all .^[1] For example, 5 × 5 persymmetric matrices are of the form

This can be equivalently expressed as where is the exchange matrix.

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

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 = (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 formA 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

Notes and References

1. . See page 193.