Persymmetric Matrix

A persymmetric matrix is a square matrix symmetric under reflection about its skew diagonal. For an matrix , this condition is

Equivalently, , where is the permutation matrix that reverses the order of coordinates and denotes the transpose. For example,

is persymmetric. Every square Toeplitz matrix is persymmetric, but a persymmetric matrix need not be a Toeplitz matrix (Petersen and Pedersen 2012).

The term has also been used as a synonym for Hankel matrix (Tamm 2001), which instead requires constant entries along each skew diagonal. A Hankel matrix need not be persymmetric in the reflection sense. For example, is a Hankel matrix but is not symmetric under reflection about its skew diagonal. The term orthosymmetric matrix is also used with these differing meanings.