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Persymmetric matrix

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Persymmetric matrix

In mathematics, persymmetric matrix may refer to:

Contents

  1. a square matrix which is symmetric in the northeast-to-southwest 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

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 for all i, j.

For example, 5-by-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 ] .

This can be equivalently expressed as AJ = JAT 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

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 = [ 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 ] .

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.

References

Persymmetric matrix Wikipedia
(Text) CC BY-SA


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