Centrosymmetric matrix

File:Matrix symmetry qtl4.svg

Symmetry pattern of a centrosymmetric 5 × 5 matrix

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

Formal definition

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

$$A_{i,j}=A_{n-i+1,n-j+1}\text{for all }i,j\in \{1,\dots ,n\}.$$

Alternatively, if Template:Mvar denotes the n × n exchange matrix with 1 on the antidiagonal and 0 elsewhere: $$J_{i,j}=\{\begin{array}{ll}1,& i+j=n+1\\ 0,& i+j\ne n+1\\ \end{array}$$ then a matrix Template:Mvar is centrosymmetric if and only if AJ = JA.

Examples

• All 2 × 2 centrosymmetric matrices have the form $$\left[\begin{array}{cc}a& b\\ b& a\end{array}\right].$$
• All 3 × 3 centrosymmetric matrices have the form $$\left[\begin{array}{ccc}a& b& c\\ d& e& d\\ c& b& a\end{array}\right].$$
• Symmetric Toeplitz matrices are centrosymmetric.

Algebraic structure and properties

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

$\frac{m^2+mmod2}{2}.$

Related structures

An n × n matrix Template:Mvar is said to be skew-centrosymmetric if its entries satisfy $$A_{i,j}=-A_{n-i+1,n-j+1}\text{for all }i,j\in \{1,\dots ,n\}.$$ Equivalently, Template:Mvar is skew-centrosymmetric if AJ = −JA, where Template:Mvar is the exchange matrix defined previously.

The centrosymmetric relation AJ = JA lends itself to a natural generalization, where Template:Mvar is replaced with an involutory matrix Template:Mvar (i.e., K² = I ) or, more generally, a matrix Template:Mvar satisfying K^m = I for an integer m > 1. The inverse problem for the commutation relation AK = KA of identifying all involutory Template:Mvar that commute with a fixed matrix Template:Mvar 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.

External links

• Centrosymmetric matrix on MathWorld.