Persymmetric matrix

{{short description|Square matrix symmetric about its anti-diagonal}}

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 == thumb|Symmetry pattern of a persymmetric 5 × 5 matrix Let {{math|1=''A'' = (''a_ij'')}} be an {{math|''n'' × ''n''}} matrix. The first definition of ''persymmetric'' requires that <math display="block">a_{ij} = a_{n-j+1,\,n-i+1}</math> for all {{math|''i'', ''j''}}.<ref>{{citation | first1=Gene H. | last1=Golub | author1-link=Gene H. Golub | first2=Charles F. | last2=Van Loan | author2-link=Charles F. Van Loan | year=1996 | title=Matrix Computations | edition=3rd | publisher=Johns Hopkins | place=Baltimore | isbn=978-0-8018-5414-9}}. See page 193.</ref> For example, 5 × 5 persymmetric matrices are of the form <math display="block">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}.</math>

This can be equivalently expressed as {{math|1=''AJ'' = ''JA''^T}} where {{mvar|J}} is the exchange matrix.

A third way to express this is seen by post-multiplying {{math|1=''AJ'' = ''JA''^T}} with {{mvar|J}} on both sides, showing that {{math|''A''^T}} rotated 180 degrees is identical to {{mvar|A}}: <math display="block">A = J A^\mathsf{T} J.</math>

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 == {{details|Hankel matrix}}

The second definition is due to Thomas Muir.<ref name="muir">{{Citation |author1-link=Thomas Muir (mathematician) |last=Muir|first=Thomas|title=Treatise on the Theory of Determinants|page= 419|publisher= Dover Press|orig-year= 1933 |isbn=978-0-486-49553-8 |oclc=52203124 |date=2003 |first2=William H. |last2=Metzler}}</ref> 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 <math display="block">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}.</math> A '''persymmetric determinant''' is the determinant of a persymmetric matrix.<ref name="muir"/>

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== {{reflist}}

{{Matrix classes}}

Category:Determinants Category:Matrices (mathematics)

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