Central symmetric matrix

Symmetry pattern of a centrally symmetric (5 × 5) matrix

In mathematics, a centrally symmetric matrix is a square matrix that is point-symmetric with respect to its center point. Centrally symmetric matrices occur, among other things, in the numerical solution of certain differential equations and in the investigation of Markov processes .

definition

A square matrix over a body is called centrally symmetric if for its entries ${\displaystyle A\in K^{n\times n}}$ ${\displaystyle K}$

${\displaystyle a_{i,j}=a_{n-i+1,n-j+1}}$

for applies. The entries of a centrally symmetric matrix do not change if they are mirrored at the center of the matrix. ${\displaystyle i,j=1,\ldots ,n}$

Examples

Centrally symmetric matrices of size have the general shape ${\displaystyle 3\times 3}$

${\displaystyle A={\begin{pmatrix}a&b&c\\d&e&d\\c&b&a\end{pmatrix}}}$

and those of size the shape ${\displaystyle 4\times 4}$

${\displaystyle A={\begin{pmatrix}a&b&c&d\\e&f&g&h\\h&g&f&e\\d&c&b&a\end{pmatrix}}}$

with . ${\displaystyle a,b,c,d,e,f,g,h\in K}$

properties

Symmetries

With the permutation matrix defined by ${\displaystyle J\in K^{n\times n}}$

${\displaystyle J=(\delta _{i,n-j+1})_{ij}={\begin{pmatrix}0&&1\\&\cdot ^{\,\cdot ^{\,\cdot }}&\\1&&0\end{pmatrix}}}$

centrally symmetric matrices can also be made compact by the condition

${\displaystyle JA=AJ}$

characterize. A centrally symmetric matrix that is also symmetric or persymmetric is called a bisymmetric matrix . Bisymmetric matrices are both as to its main diagonal , as well as with regard to their anti-diagonal symmetrical.

Block structure

Centrally symmetric matrices with an even number of rows and columns have a special block structure of the form

${\displaystyle A={\begin{bmatrix}B&JCJ\\C&JBJ\end{bmatrix}}\in K^{2n\times 2n}}$,

where are. Centrally symmetric matrices with an odd number of rows and columns have the structure ${\displaystyle B,C\in K^{n\times n}}$

${\displaystyle A={\begin{bmatrix}B&y&JCJ\\x&z&xJ\\C&Jy&JBJ\end{bmatrix}}\in K^{(2n+1)\times (2n+1)}}$,

wherein , , and are. ${\displaystyle B,C\in K^{n\times n}}$ ${\displaystyle x\in K^{1\times n}}$ ${\displaystyle y\in K^{n\times 1}}$ ${\displaystyle z\in K}$

Eigenvalues

The eigenvalues ​​of a centrally symmetric matrix with an even number of rows and columns are then given as the eigenvalues ​​of the matrices

${\displaystyle F=B-JC}$   and   . ${\displaystyle G=B+JC}$

The corresponding eigenvectors then have the form

${\displaystyle {\begin{pmatrix}u\\-Ju\end{pmatrix}}}$   and   , ${\displaystyle {\begin{pmatrix}v\\Jv\end{pmatrix}}}$

where is an eigenvector of and an eigenvector of . For centrally symmetric matrices with an odd number of rows and columns, the eigenvalues ​​are given as the eigenvalues ​​of the matrices ${\displaystyle u}$ ${\displaystyle F}$ ${\displaystyle v}$ ${\displaystyle G}$

${\displaystyle F=B-JC}$   and   . ${\displaystyle {\bar {G}}={\begin{bmatrix}z&{\sqrt {2}}x\\{\sqrt {2}}y&B+JC\end{bmatrix}}}$

The corresponding eigenvectors then have the form

${\displaystyle {\begin{pmatrix}u\\0\\-Ju\end{pmatrix}}}$   and   , ${\displaystyle {\begin{pmatrix}{\bar {v}}\\{\sqrt {2}}\alpha \\J{\bar {v}}\end{pmatrix}}}$

where is an eigenvector of and an eigenvector of . ${\displaystyle u}$ ${\displaystyle F}$ ${\displaystyle (\alpha ,{\bar {v}}^{T})^{T}}$ ${\displaystyle {\bar {G}}}$

Sum and product

The sum of two centrally symmetric matrices and again results in a centrally symmetric matrix, also scalar multiples are with . Since the zero matrix is trivially symmetric, the symmetric matrices form a sub-vector space in the matrix space . ${\displaystyle A+B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle cA}$ ${\displaystyle c\in K}$ ${\displaystyle K^{n\times n}}$

The product of two centrally symmetric matrices also results in a centrally symmetric matrix, because it holds ${\displaystyle A\cdot B}$

${\displaystyle JAB=AJB=ABJ}$.

Since the identity matrix is also centrally symmetric, the centrally symmetric matrices form a subalgebra of the associative algebra of the square matrices.

Inverse

The inverse of a regular centrally symmetric matrix is ​​again centrally symmetric because it holds ${\displaystyle A^{-1}}$

${\displaystyle JA^{-1}=(AJ)^{-1}=(JA)^{-1}=A^{-1}J}$.

The regular centrally symmetric matrices thus form a subgroup of the general linear group . ${\displaystyle \mathrm {GL} (n,K)}$

Applications

Centrally symmetric matrices occur, for example, in the numerical solution of certain differential equations and eigenvalue problems, in the investigation of Markov processes and in a number of physical problems.

See also

• Hankel matrix
• Toeplitz matrix

literature

• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 978-0-521-83940-2 .

Individual evidence

1. ^ Thomas Muir: A Treatise on the Theory of Determinants . Dover, New York 1960, pp. 19 .
2. ^ ^{A b} Roger A. Horn, Charles Johnson: Matrix analysis . Cambridge University Press, 2013, pp. 36 .
3. ↑ Iyad T. Abu-Jeib: Centrosymmetric matrices: Properties and on Alternative Approach . In: Canadian Applied Mathematics Quarterly . tape 10 , no. 4 , 2002, p. 431 .
4. ↑ Alan L. Andrew: Eigenvectors of Certain matrices . In: Linear Algebra and Applications . No. 7 , 1973, p. 157-162 .
5. ↑ James R. Weaver: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors . In: American Mathematical Monthly . No. 92 , 1985, pp. 711-717 .
6. ^ Lokesh Datta, Salvatore D. Morgera: On the reducibility of centrosymmetric matrices — applications in engineering problems . In: Circuits Systems and Signal Processing . No. 8 , 1989, pp. 71-96 .

Web links

• Eric W. Weisstein : Centrosymmetric Matrix . In: MathWorld (English).