In mathematics, persymmetric matrix may refer to: a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or 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 for all .. See page 193. 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 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 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
References