In mathematics, persymmetric matrix may refer to:
- a square matrix which is symmetric in the northeast-to-southwest 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.
Let A = (aij) be an n × n matrix. The first definition of persymmetric requires that
- for all i, j.
For example, 5-by-5 persymmetric matrices are of the form
This can be equivalently expressed as AJ = JAT 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.
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 matrix for which the values on each line parallel to the main diagonal are constant, is called a Toeplitz matrix.