Persymmetric

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In mathematics, persymmetric matrix may refer to:

  1. a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or
  2. 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

Symmetry pattern of a persymmetric 5 × 5 matrix

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}} {\displaystyle a_{ij}=a_{n-j+1,\,n-i+1}} for all i, j.[1] 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}}.} {\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.} {\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.[2] 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 = [ r 1 r 2 r 3 ⋯ r n r 2 r 3 r 4 ⋯ r n + 1 r 3 r 4 r 5 ⋯ r n + 2 ⋮ ⋮ ⋮ ⋱ ⋮ r n r n + 1 r n + 2 ⋯ r 2 n − 1 ] . {\displaystyle 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}}.} {\displaystyle 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.

See also

References

  1. 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. Muir, Thomas; Metzler, William H. (2003) [1933], Treatise on the Theory of Determinants, Dover Press, p. 419, ISBN 978-0-486-49553-8, OCLC 52203124