Persymmetric matrix

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In mathematics, persymmetric matrix may refer to:

a square matrix which is symmetric in 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.

Definition 1

Let A = (a_i⁣j) be an n × n matrix. The first definition of persymmetric requires that

$a_{{ij}}=a_{{n-j+1,n-i+1}}$ for all i, j.^[1]

For example, 5-by-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 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.

Definition 2

For more details on this topic, see Hankel matrix.

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

$A={\begin{bmatrix}r_{1}&r_{2}&r_{3}&\cdots &r_{n}\\r_{2}&r_{3}&r_{4}&\cdots &r_{{n+1}}\\r_{3}&r_{4}&r_{5}&\cdots &r_{{n+2}}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n}&r_{{n+1}}&r_{{n+2}}&\cdots &r_{{2n-1}}\end{bmatrix}}.$

A persymmetric determinant is the determinant of a persymmetric matrix.^[2]

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

References

↑ Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9 . See page 193.
↑ 2.0 2.1 Muir, Thomas (1960), Treatise on the Theory of Determinants, Dover Press, p. 419

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