Persymmetric matrix

Let be an matrix. The first definition of persymmetric requires that a_{ij} = a_{n-j+1,\,n-i+1} for all . For example, 5 × 5 persymmetric matrices are of the form 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 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 = J A^\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 A = \begin{bmatrix} r_1 & r_2 & r_3 & \cdots & r_n \\ r_2 & r_3 & r_4 & \cdots & r_{n+1} \\ r_3 & r_4 & r_5 & \cdots & r_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1} \end{bmatrix}. 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==