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| In [[mathematics]], '''persymmetric matrix''' may refer to:
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| # a [[square matrix]] which is symmetric in the northeast-to-southwest diagonal; or
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| # a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.
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| 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.
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| == Definition 1 ==
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| Let ''A'' = (''a''<sub>''i''''j''</sub>) be an ''n'' × ''n'' matrix. The first definition of ''persymmetric'' requires that
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| :<math> a_{ij} = a_{n-j+1,n-i+1} </math> for all ''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-by-5 persymmetric matrices are of the form
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| :<math> A = \begin{bmatrix} | |
| a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
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| a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\
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| a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\
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| a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\
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| a_{51} & a_{41} & a_{31} & a_{21} & a_{11}
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| \end{bmatrix}. </math>
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| This can be equivalently expressed as ''AJ = JA''<sup>T</sup> where ''J'' is the [[exchange matrix]].
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| 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 matrix|bisymmetric matrices]].
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| == Definition 2 == | |
| {{details|Hankel matrix}}
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| The second definition is due to [[Thomas Muir (mathematician)|Thomas Muir]].<ref name="muir">{{Citation|last=Muir|first=Thomas|title=Treatise on the Theory of Determinants|page= 419|publisher= Dover Press|year= 1960}}</ref> It says that the square matrix ''A'' = (''a''<sub>''ij''</sub>) is persymmetric if ''a''<sub>''ij''</sub> depends only on ''i'' + ''j''. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
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| :<math> A = \begin{bmatrix}
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| r_1 & r_2 & r_3 & \cdots & r_n \\
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| r_2 & r_3 & r_4 & \cdots & r_{n+1} \\
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| r_3 & r_4 & r_5 & \cdots & r_{n+2} \\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1}
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| \end{bmatrix}.
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| </math>
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| A '''persymmetric determinant''' is the [[determinant]] of a persymmetric matrix.<ref name="muir"/>
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| A matrix for which the values on each line parallel to the main diagonal are constant, is called a [[Toeplitz matrix]].
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| ==References==
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| {{Reflist}}
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| [[Category:Determinants]]
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| [[Category:Matrices]]
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