Persymmetric matrix

In mathematics, persymmetric matrix may refer to:
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 "A" = ("a""i&hairsp;j") be an "n" × "n" matrix. The first definition of "persymmetric" requires that 
For example, 5 × 5 persymmetric matrices are of the form
This can be equivalently expressed as "AJ" = "JA"T 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.
Definition 2.
The second definition is due to Thomas Muir. It says that the square matrix "A" = ("a""i&hairsp;j") is persymmetric if "a""i&hairsp;j" 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.