persymmetric

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 be an matrix. The first definition of ''persymmetric'' requires that
$a_ = a_$ for all .. See page 193.
For example, 5 × 5 persymmetric matrices are of the form
$A = \begin a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \end.$

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 :
$A = J A^\mathsf J.$

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'' = (''a''_''ij'') is persymmetric if ''a''_''ij'' depends only on ''i'' + ''j''. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
$A = \begin r_1 & r_2 & r_3 & \cdots & r_n \\ r_2 & r_3 & r_4 & \cdots & r_ \\ r_3 & r_4 & r_5 & \cdots & r_ \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_n & r_ & r_ & \cdots & r_ \end.$
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

* Centrosymmetric matrix

References

Determinants
Matrices (mathematics)

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