Centrosymmetric matrix explained

In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center.

Formal definition

An matrix is centrosymmetric when its entries satisfy

$$A\_\; =\; A\_\; \backslash quad\; \backslash texti,j\; \backslash in\; \backslash .$$

Alternatively, if denotes the exchange matrix with 1 on the antidiagonal and 0 elsewhere: then a matrix is centrosymmetric if and only if .

Examples

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

\begin a & b \\ b & a \end.

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

\begin a & b & c \\ d & e & d \\ c & b & a \end.

• Symmetric Toeplitz matrices are centrosymmetric.

Algebraic structure and properties

• If and are centrosymmetric matrices over a field, then so are and for any in . Moreover, the matrix product is centrosymmetric, since . Since the identity matrix is also centrosymmetric, it follows that the set of centrosymmetric matrices over forms a subalgebra of the associative algebra of all matrices.
• If is a centrosymmetric matrix with an -dimensional eigenbasis, then its eigenvectors can each be chosen so that they satisfy either or where is the exchange matrix.
• If is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with must be centrosymmetric.
• The maximum number of unique elements in an centrosymmetric matrix is

(m ^{2} + m \bmod 2)/(2) .

Related structures

An matrix is said to be skew-centrosymmetric if its entries satisfy $$A\_\; =\; -A\_\; \backslash quad\; \backslash texti,j\; \backslash in\; \backslash .$$Equivalently, is skew-centrosymmetric if, where is the exchange matrix defined previously.

The centrosymmetric relation lends itself to a natural generalization, where is replaced with an involutory matrix (i.e.,)^{[1] [2] [3]} or, more generally, a matrix satisfying for an integer .^[4] The inverse problem for the commutation relation of identifying all involutory that commute with a fixed matrix has also been studied.^[4]

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.^[2] A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.^[5]

Further reading

• Book: Muir, Thomas. Thomas Muir (mathematician). 1960. A Treatise on the Theory of Determinants. registration. Dover. 19. 0-486-60670-8.
• 10.2307/2323222. James R. . Weaver. Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. American Mathematical Monthly. 92. 10. 1985. 711–717. 2323222 .

External links

• Centrosymmetric matrix on MathWorld.

Notes and References

1. 10.1016/0024-3795(73)90049-9. Alan . Andrew. Eigenvectors of certain matrices. Linear Algebra Appl.. 7 . 1973. 2. 151–162. free.
2. Tao . David . Yasuda . Mark . A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices . SIAM J. Matrix Anal. Appl. . 23 . 3 . 885–895 . 2002 . 10.1137/S0895479801386730 .
3. 10.1016/j.laa.2003.07.013. W. F.. Trench. Characterization and properties of matrices with generalized symmetry or skew symmetry. Linear Algebra Appl. . 377 . 2004. 207–218. free.
4. Yasuda . Mark . Some properties of commuting and anti-commuting m-involutions . Acta Mathematica Scientia . 32 . 2 . 631–644 . 2012. 10.1016/S0252-9602(12)60044-7.
5. Yasuda . Mark . A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices . SIAM J. Matrix Anal. Appl. . 25 . 3 . 601–605 . 2003 . 10.1137/S0895479802418835.