Bisymmetric matrix

{{Short description|Square matrix symmetric about both its diagonal and anti-diagonal}}

In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an {{math|n × n}} matrix {{mvar|A}} is bisymmetric if it satisfies both {{math|1=A = A^T}} (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

$\begin{bmatrix}
a & b & c & d & e \\
b & f & g & h & d \\
c & g & i & g & c \\
d & h & g & f & b \\
e & d & c & b & a \end{bmatrix}
= \begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{12} & a_{22} & a_{23} & a_{24} & a_{14} \\
a_{13} & a_{23} & a_{33} & a_{23} & a_{13} \\
a_{14} & a_{24} & a_{23} & a_{22} & a_{12} \\
a_{15} & a_{14} & a_{13} & a_{12} & a_{11}
\end{bmatrix}$

is bisymmetric. The associated $5\times 5$ exchange matrix for this example is

$J_{5} = \begin{bmatrix}
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0
\end{bmatrix}$

Properties

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.{{cite journal

|last=Tao

|first=David

|author2=Yasuda, Mark

|title=A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices

|journal=SIAM Journal on Matrix Analysis and Applications

|volume=23

|issue=3

|pages=885–895

|year=2002

|doi=10.1137/S0895479801386730

|url=https://zenodo.org/record/1236140

}}

If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.{{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}}
The inverse of bisymmetric matrices can be represented by recurrence formulas.{{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}}

References

Category:Matrices (mathematics)