Persymmetric matrix

Symmetry pattern of a persymmetric (5 × 5) matrix

In mathematics, a persymmetric matrix is a square matrix that is symmetric with respect to its counter-diagonal .

definition

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

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

for applies. The entries of a persymmetrical matrix do not change if they are mirrored on the opposite diagonal. ${\displaystyle i,j=1,\ldots ,n}$

Examples

For example, a real persymmetric matrix of size is ${\displaystyle 3\times 3}$

${\displaystyle A={\begin{pmatrix}1&2&3\\2&3&2\\3&2&1\end{pmatrix}}}$

Generally, persymmetric matrices of size have the form ${\displaystyle 3\times 3}$

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

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

properties

Symmetries

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

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

Persymmetric matrices can also be made compact by the condition

${\displaystyle JA=A^{T}J}$

characterize. A bisymmetrical matrix is a persymmetrical matrix that is also symmetrical or centrally symmetrical . A Toeplitz matrix is a persymmetrical matrix whose entries on the main diagonal and all secondary diagonals are constant. A cyclic matrix is a persymmetric matrix whose entries are constant on all diagonals and are repeated cyclically .

Sum and product

The sum of two persymmetric matrices and again results in a persymmetric matrix, as are scalar multiples with . Since the zero matrix is trivially persymmetric, the persymmetric 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 persymmetric matrices results from ${\displaystyle A\cdot B}$

${\displaystyle JAB=A^{T}JB=A^{T}B^{T}J=(BA)^{T}J}$

a persymmetric matrix again if and only if the two matrices and commute . ${\displaystyle A}$ ${\displaystyle B}$

Inverse

For the inverse of a persymmetric matrix, if it exists ${\displaystyle A^{-1}}$

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

The inverse of a regular persymmetric matrix is ​​therefore again persymmetric.

See also

• Hankel matrix

literature

• Gene Golub, Charles van Loan: Matrix Computations . JHU Press, 2013, ISBN 978-1-4214-0794-4 .
• Martin Hanke-Bourgeois: Fundamentals of Numerical Mathematics and Scientific Computing . Springer, 2008, ISBN 978-3-8348-0708-3 .
• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 978-0-521-83940-2 .

Individual evidence

1. ↑ Martin Hanke-Bourgeois: Fundamentals of numerical mathematics and scientific computing . Springer, 2008, p. 66 .
2. ^ Roger A. Horn, Charles Johnson: Matrix analysis . Cambridge University Press, 2013, pp. 36 .
3. ^ Gene Golub, Charles van Loan: Matrix Computations . JHU Press, 2013, p. 208 .