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| In [[mathematics]], especially in [[linear algebra]] and [[Matrix (mathematics)|matrix theory]], a '''centrosymmetric matrix''' is a [[matrix (mathematics)|matrix]] which is symmetric about its center. More precisely, an ''n'' × ''n'' matrix ''A'' = [ ''A''<sub>i,j</sub> ] is centrosymmetric when its entries satisfy
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| :''A''<sub>i,j</sub> = ''A''<sub>n−i+1,n−j+1</sub> for 1 ≤ i,j ≤ n. | |
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| If ''J'' denotes the ''n'' × ''n'' matrix with 1 on the counterdiagonal and 0 elsewhere (that is, ''J''<sub>i,n+1-i</sub> = 1; ''J''<sub>i,j</sub> = 0 if j ≠ n+1-i), then a matrix ''A'' is centrosymmetric if and only if ''AJ = JA''. The matrix ''J'' is sometimes referred to as the [[exchange matrix]].
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| ==Examples==
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| * All 2×2 centrosymmetric matrices have the form
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| :<math> \begin{bmatrix} a & b \\ b & a \end{bmatrix}. </math> | |
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| * All 3×3 centrosymmetric matrices have the form
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| :<math> \begin{bmatrix} a & b & c \\ d & e & d \\ c & b & a \end{bmatrix}. </math>
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| * [[Symmetric matrix|Symmetric]] [[Toeplitz matrix|Toeplitz]] matrices are centrosymmetric.
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| ==Algebraic structure==
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| If ''A'' and ''B'' are centrosymmetric matrices over a given [[field (mathematics)|field]] ''K'', then so are ''A+B'' and ''cA'' for any ''c'' in ''K''. In addition, the [[matrix product]] ''AB'' is centrosymmetric, since ''JAB = AJB = ABJ''. Since the [[identity matrix]] is also centrosymmetric, it follows that the set of ''n × n'' centrosymmetric matrices over ''K'' is a subalgebra of the [[associative algebra]] of all ''n × n'' matrices.
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| ==Related structures==
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| An ''n × n'' matrix ''A'' is said to be ''skew-centrosymmetric'' if its entries satisfy ''A''<sub>i,j</sub> = -''A''<sub>n−i+1,n−j+1</sub> for 1 ≤ i,j ≤ n. Equivalently, ''A'' is skew-centrosymmetric if ''AJ = -JA'', where ''J'' is the exchange matrix defined above.
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| The centrosymmetric relation ''AJ = JA'' lends itself to a natural generalization, where ''J'' is replaced with an [[involutary matrix]] ''K'' (i.e., ''K<sup>2</sup> = I'')<ref name="AA">{{Cite journal|doi=10.1016/0024-3795(73)90049-9|first=A. |last=Andrew|title= Eigenvectors of certain matrices|journal= Linear Algebra Appl.| volume= 7 |year=1973|issue=2|pages=151–162|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| <ref name="simax0">{{cite journal | last1 = Tao | first1 = D. | last2 = Yasuda|first2=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 | year = 2002 | url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000023000003000885000001&idtype=cvips&gifs=Yes
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| | accessdate = 2007-10-12 | doi = 10.1137/S0895479801386730}}</ref>
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| <ref name="laa">{{cite journal|doi=10.1016/j.laa.2003.07.013|first=W. F.|last= Trench|title= Characterization and properties of matrices with generalized symmetry or skew symmetry|journal=Linear Algebra Appl. |volume=377 |year=2004|pages=207–218}}</ref> or, more generally, a matrix ''K'' satisfying ''K<sup>m</sup> = I'' for an integer ''m > 1''.<ref name=acta>{{cite journal | last = Yasuda | first = M. | title = Some properties of commuting and anti-commuting m-involutions | journal = Acta Mathematica Scientia | volume = 32 | issue = 2 | pages = 631–644 | year = 2012 | url = http://www.sciencedirect.com/science/article/pii/S0252960212600447 | accessdate = 2013-02-18| doi = 10.1016/S0252-9602(12)60044-7}}</ref>
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| [[Symmetric matrix|Symmetric]] centrosymmetric matrices are sometimes called [[bisymmetric matrix|bisymmetric matrices]]. When the [[Field (mathematics)|ground field]] is the field of [[real numbers]], it has been shown that 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"/> A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.<ref name="simax1">{{cite journal | last = Yasuda | first = M. | title = A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 25 | issue = 3 | pages = 601–605 | year = 2003 | url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000025000003000601000001&idtype=cvips&gifs=Yes
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| | accessdate = 2007-10-12 | doi = 10.1137/S0895479802418835}}</ref>
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{cite book|first=Thomas|last=Muir|author-link=Thomas Muir (mathematician)|year=1960|title=A Treatise on the Theory of Determinants|publisher=Dover|page=19|isbn= 0-486-60670-8}}
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| * {{cite journal|doi=10.2307/2323222|first=J. R. |last=Weaver|title= Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors|journal=American Mathematical Monthly|volume=92|issue=10|year=1985|pages=711–717}}
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| ==External links==
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| * [http://mathworld.wolfram.com/CentrosymmetricMatrix.html Centrosymmetric matrix] on [[MathWorld]].
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| {{DEFAULTSORT:Centrosymmetric Matrix}}
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| [[Category:Linear algebra]]
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| [[Category:Matrices]]
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