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 A = (aij) be an n × n matrix. The first definition of persymmetric requires that
a i j = a n − j + 1 , n − i + 1 {\displaystyle a_{ij}=a_{n-j+1,\,n-i+1}} for all i, j. For example, 5 × 5 persymmetric matrices are of the form
A = [ a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 14 a 31 a 32 a 33 a 23 a 13 a 41 a 42 a 32 a 22 a 12 a 51 a 41 a 31 a 21 a 11 ] . {\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\a_{21}&a_{22}&a_{23}&a_{24}&a_{14}\\a_{31}&a_{32}&a_{33}&a_{23}&a_{13}\\a_{41}&a_{42}&a_{32}&a_{22}&a_{12}\\a_{51}&a_{41}&a_{31}&a_{21}&a_{11}\end{bmatrix}}.}
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 = J A T J . {\displaystyle A=JA^{\mathsf {T}}J.}
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
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