Bisymmetric matrix: Difference between revisions
Content deleted Content added
m Double-checking that a bug has been fixed. You can use this bot yourself! Please report any bugs. |
put in an example |
||
| Line 1: | Line 1: | ||
In [[mathematics]], a '''bisymmetric matrix''' is a square [[matrix (mathematics)|matrix]] that is symmetric about both of its main diagonals. More precisely, an ''n'' × ''n'' matrix ''A'' is bisymmetric if and only if it satisfies ''A = A<sup>T</sup>'' and ''AJ = JA'' where ''J'' is the ''n'' × ''n'' [[exchange matrix]]. |
In [[mathematics]], a '''bisymmetric matrix''' is a square [[matrix (mathematics)|matrix]] that is symmetric about both of its main diagonals. More precisely, an ''n'' × ''n'' matrix ''A'' is bisymmetric if and only if it satisfies ''A = A<sup>T</sup>'' and ''AJ = JA'' where ''J'' is the ''n'' × ''n'' [[exchange matrix]]. |
||
For example: |
|||
:<math>\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}.</math> |
|||
==Properties== |
==Properties== |
||
Bisymmetric matrices are both symmetric [[centrosymmetric matrix|centrosymmetric]] and symmetric [[persymmetric matrix|persymmetric]]. It has been shown that real-valued bisymmetric matrices are precisely those symmetric matrices whose [[eigenvalues]] are the same up to sign after pre or post multiplication by the exchange matrix<ref name="simax0">{{cite journal | last = Tao | first = D. | coauthors = Yasuda, M. | title = A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 23 | issue = 3 | pages = 885–895 | date = 2002 | url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000023000003000885000001&idtype=cvips&gifs=Yes |
Bisymmetric matrices are both symmetric [[centrosymmetric matrix|centrosymmetric]] and symmetric [[persymmetric matrix|persymmetric]]. It has been shown that real-valued bisymmetric matrices are precisely those symmetric matrices whose [[eigenvalues]] are the same up to sign after pre or post multiplication by the exchange matrix<ref name="simax0">{{cite journal | last = Tao | first = D. | coauthors = Yasuda, M. | title = A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 23 | issue = 3 | pages = 885–895 | date = 2002 | url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000023000003000885000001&idtype=cvips&gifs=Yes |
||
| accessdate = 2007-10-12 | doi = 10.1137/S0895479801386730}}</ref>. |
| accessdate = 2007-10-12 | doi = 10.1137/S0895479801386730}}</ref>. |
||
The product of two bisymmetric matrices results in a centrosymetric matrix |
|||
==References== |
==References== |
||
Revision as of 23:05, 30 October 2009
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 and only if it satisfies A = AT and AJ = JA where J is the n × n exchange matrix.
For example:
Properties
Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric. It has been shown that real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues are the same up to sign after pre or post multiplication by the exchange matrix[1].
The product of two bisymmetric matrices results in a centrosymetric matrix
References
- ^ Tao, D. (2002). "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM J. Matrix Anal. Appl. 23 (3): 885–895. doi:10.1137/S0895479801386730. Retrieved 2007-10-12.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help)