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{{Short description|Square matrix symmetric about both its diagonal and anti-diagonal}}
[[File:Matrix symmetry qtl3.svg|thumb|Symmetry pattern of a bisymmetric 5 × 5 matrix]]
[[File:Matrix symmetry qtl3.svg|thumb|Symmetry pattern of a bisymmetric 5 × 5 matrix]]

In [[mathematics]], a '''bisymmetric matrix''' is a [[square matrix]] that is symmetric about both of its main diagonals. More precisely, an ''n''&thinsp;×&thinsp;''n'' matrix ''A'' is bisymmetric if it satisfies both ''A'' = ''A<sup>T</sup>'' and ''AJ'' = ''JA'' where ''J'' is the ''n''&thinsp;×&thinsp;''n'' [[exchange matrix]].
In [[mathematics]], a '''bisymmetric matrix''' is a [[square matrix]] that is [[Symmetric matrix|symmetric]] about both of its main [[Diagonal of a matrix|diagonals]]. More precisely, an {{math|''n'' × ''n''}} matrix {{mvar|A}} is bisymmetric if it satisfies both {{math|1=''A'' = ''A''<sup>T</sup>}} (it is its own [[transpose]]), and {{math|1=''AJ'' = ''JA''}}, where {{mvar|J}} is the {{math|''n'' × ''n''}} [[exchange matrix]].


For example, any matrix of the form
For example, any matrix of the form


:<math>\begin{bmatrix}
<math display=block>\begin{bmatrix}
a & b & c & d & e \\
a & b & c & d & e \\
b & f & g & h & d \\
b & f & g & h & d \\
c & g & i & g & c \\
c & g & i & g & c \\
d & h & g & f & b \\
d & h & g & f & b \\
e & d & c & b & a \end{bmatrix}</math>
e & d & c & b & a \end{bmatrix}
= \begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{12} & a_{22} & a_{23} & a_{24} & a_{14} \\
a_{13} & a_{23} & a_{33} & a_{23} & a_{13} \\
a_{14} & a_{24} & a_{23} & a_{22} & a_{12} \\
a_{15} & a_{14} & a_{13} & a_{12} & a_{11}
\end{bmatrix}</math>


is bisymmetric. The associated <math>5\times 5</math> [[exchange matrix]] for this example is
is bisymmetric.

<math>J_{5} = \begin{bmatrix}
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0
\end{bmatrix}</math>


==Properties==
==Properties==
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}}</ref>
}}</ref>
*If ''A'' is a real bisymmetric matrix with distinct eigenvalues, then the matrices that [[commuting matrices|commute]] with ''A'' must be bisymmetric.<ref name=acta>{{cite journal | last = Yasuda | first = Mark | title = Some properties of commuting and anti-commuting m-involutions | journal = Acta Mathematica Scientia | volume = 32 | issue = 2 | pages = 631–644 | year = 2012| doi = 10.1016/S0252-9602(12)60044-7}}</ref>
*If ''A'' is a real bisymmetric matrix with distinct eigenvalues, then the matrices that [[commuting matrices|commute]] with ''A'' must be bisymmetric.<ref name=acta>{{cite journal | last = Yasuda | first = Mark | title = Some properties of commuting and anti-commuting m-involutions | journal = Acta Mathematica Scientia | volume = 32 | issue = 2 | pages = 631–644 | year = 2012| doi = 10.1016/S0252-9602(12)60044-7}}</ref>
*The [[inverse matrix|inverse]] of bisymmetric matrices can be represented by recurrence formulas.<ref>{{Cite journal|last=Wang|first=Yanfeng|last2=Lü|first2=Feng|last3=Lü|first3=Weiran|date=2018-01-10|title=The inverse of bisymmetric matrices|journal=Linear and Multilinear Algebra|volume=0|issue=3|pages=479–489|doi=10.1080/03081087.2017.1422688|issn=0308-1087}}</ref>
*The [[inverse matrix|inverse]] of bisymmetric matrices can be represented by recurrence formulas.<ref>{{Cite journal|last1=Wang|first1=Yanfeng|last2=Lü|first2=Feng|last3=Lü|first3=Weiran|date=2018-01-10|title=The inverse of bisymmetric matrices|journal=Linear and Multilinear Algebra|volume=67|issue=3|pages=479–489|doi=10.1080/03081087.2017.1422688|s2cid=125163794|issn=0308-1087}}</ref>


==References==
==References==
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{{DEFAULTSORT:Bisymmetric Matrix}}
{{DEFAULTSORT:Bisymmetric Matrix}}
[[Category:Matrices]]
[[Category:Matrices]]


{{matrix-stub}}

Latest revision as of 01:22, 2 December 2023

Symmetry pattern of a bisymmetric 5 × 5 matrix

In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT (it is its own transpose), and AJ = JA, where J is the n × n exchange matrix.

For example, any matrix of the form

is bisymmetric. The associated exchange matrix for this example is

Properties

[edit]
  • Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.
  • The product of two bisymmetric matrices is a centrosymmetric matrix.
  • Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.[1]
  • If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.[2]
  • The inverse of bisymmetric matrices can be represented by recurrence formulas.[3]

References

[edit]
  1. ^ Tao, David; Yasuda, Mark (2002). "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM Journal on Matrix Analysis and Applications. 23 (3): 885–895. doi:10.1137/S0895479801386730.
  2. ^ Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia. 32 (2): 631–644. doi:10.1016/S0252-9602(12)60044-7.
  3. ^ Wang, Yanfeng; Lü, Feng; Lü, Weiran (2018-01-10). "The inverse of bisymmetric matrices". Linear and Multilinear Algebra. 67 (3): 479–489. doi:10.1080/03081087.2017.1422688. ISSN 0308-1087. S2CID 125163794.