Centrosymmetric matrix

An {{math|n × n}} matrix {{math|1=A = [A_{i, j}]}} is centrosymmetric when its entries satisfy

$$A\_\{i,\backslash ,j\}\; =\; A\_\{n-i+1,\backslash ,n-j+1\}\; \backslash quad\; \backslash text\{for\; all\; \}i,j\; \backslash in\; \backslash \{1,\backslash ,\; \backslash ldots,\backslash ,\; n\backslash \}.$$

Alternatively, if {{mvar|J}} denotes the {{math|n × n}} exchange matrix with 1 on the antidiagonal and 0 elsewhere:

$$J\_\{i,\backslash ,j\}\; =\; \backslash begin\{cases\}$$

1, & i + j = n + 1 \\

0, & i + j \ne n + 1\\

\end{cases}

then a matrix {{mvar|A}} is centrosymmetric if and only if {{math|1=AJ = JA}}.

• All 2 × 2 centrosymmetric matrices have the form $$$$

\begin{bmatrix} a & b \\ b & a \end{bmatrix}.

• All 3 × 3 centrosymmetric matrices have the form $$$$

\begin{bmatrix} a & b & c \\ d & e & d \\ c & b & a \end{bmatrix}.

• Symmetric Toeplitz matrices are centrosymmetric.

• If {{mvar|A}} and {{mvar|B}} are {{math|n × n}} centrosymmetric matrices over a field {{mvar|F}}, then so are {{math|A + B}} and {{mvar|cA}} for any {{mvar|c}} in {{mvar|F}}. Moreover, the matrix product {{mvar|AB}} is centrosymmetric, since {{math|1=JAB = AJB = ABJ}}. Since the identity matrix is also centrosymmetric, it follows that the set of {{math|n × n}} centrosymmetric matrices over {{mvar|F}} forms a subalgebra of the associative algebra of all {{math|n × n}} matrices.
• If {{mvar|A}} is a centrosymmetric matrix with an {{mvar|m}}-dimensional eigenbasis, then its {{mvar|m}} eigenvectors can each be chosen so that they satisfy either {{math|1=x = J x}} or {{math|1=x = − J x}} where {{mvar|J}} is the exchange matrix.
• If {{mvar|A}} is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with {{mvar|A}} must be centrosymmetric.
• The maximum number of unique elements in an {{math|m × m}} centrosymmetric matrix is

::$\backslash frac\{m^2\; +\; m\; \backslash bmod\; 2\}\{2\}.$

An {{math|n × n}} matrix {{mvar|A}} is said to be skew-centrosymmetric if its entries satisfy

$$A\_\{i,\backslash ,j\}\; =\; -A\_\{n-i+1,\backslash ,n-j+1\}\; \backslash quad\; \backslash text\{for\; all\; \}i,j\; \backslash in\; \backslash \{1,\backslash ,\; \backslash ldots,\backslash ,\; n\backslash \}.$$

Equivalently, {{mvar|A}} is skew-centrosymmetric if {{math|1=AJ = −JA}}, where {{mvar|J}} is the exchange matrix defined previously.

The centrosymmetric relation {{math|1=AJ = JA}} lends itself to a natural generalization, where {{mvar|J}} is replaced with an involutory matrix {{mvar|K}} (i.e., {{math|1=K² = I }}){{Cite journal|doi=10.1016/0024-3795(73)90049-9|first=Alan |last=Andrew|title= Eigenvectors of certain matrices|journal= Linear Algebra Appl.| volume= 7 |year=1973|issue=2|pages=151–162|doi-access=free}}{{cite journal

|last1=Tao

|first1=David

|last2=Yasuda

|first2=Mark

|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

|doi=10.1137/S0895479801386730

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

}}{{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|doi-access=free}} or, more generally, a matrix {{mvar|K}} satisfying {{math|1=K^m = I}} for an integer {{math|m > 1}}.{{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 problem for the commutation relation {{math|1=AK = KA}} of identifying all involutory {{mvar|K}} that commute with a fixed matrix {{mvar|A}} has also been studied.

Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the real numbers, it has been shown that 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. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.{{cite journal | last = Yasuda | first = Mark | 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 | doi = 10.1137/S0895479802418835}}

{{reflist}}

• {{cite book|first=Thomas|last=Muir|author-link=Thomas Muir (mathematician)|year=1960|title=A Treatise on the Theory of Determinants|url=https://archive.org/details/treatiseontheory0000muir|url-access=registration|publisher=Dover|page=[https://archive.org/details/treatiseontheory0000muir/page/19 19]|isbn= 0-486-60670-8}}
• {{cite journal|doi=10.2307/2323222|first=James 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|jstor=2323222 }}

• [http://mathworld.wolfram.com/CentrosymmetricMatrix.html Centrosymmetric matrix] on MathWorld.

{{Matrix classes}}

{{DEFAULTSORT:Centrosymmetric Matrix}}

Category:Linear algebra

Category:Matrices (mathematics)