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## Definition 1

## Definition 2

## See also

## References

In mathematics, **persymmetric matrix** may refer to:

- a square matrix which is symmetric with respect to the northeast-to-southwest 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.

Let *A* = (*a*_{ij}) be an *n* × *n* matrix. The first definition of *persymmetric* requires that

- for all
*i*,*j*.^{[1]}

For example, 5-by-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.

The second definition is due to Thomas Muir.^{[2]} 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 **persymmetric determinant** is the determinant of a persymmetric matrix.^{[2]}

A matrix for which the values on each line parallel to the main diagonal are constant, is called a Toeplitz matrix.

**^**Golub, Gene H.; Van Loan, Charles F. (1996),*Matrix Computations*(3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9. See page 193.- ^
^{a}^{b}Muir, Thomas (1960),*Treatise on the Theory of Determinants*, Dover Press, p. 419

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