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

for all i, j.[1]

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

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:

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.[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

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.

See also

References

  1. ^ Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9. See page 193.
  2. ^ a b Muir, Thomas (1960), Treatise on the Theory of Determinants, Dover Press, p. 419