Different kinds of symmetry for matrices We can define symmetry in real matrices in different way. However the most important one, it seems,  is this kind of symmetry linked with the main diagonal.
Beside this the most known kind of symmetry (where $A=A^T$) we could define also so called  Persymmetric matrices  (linked with antidiagonal) or Centrosymmetric matrices (linked with central entry of the matrix), however these kinds of symmetry are much less used, but also interesting for their regularity.  
It seems that the main theorem with the use of symmetric matrices i.e. that every matrix can be decomposed in an unique way into symmetric and skew-symmetric part ($A= sym(A)+sk(A)$)  could have probably their cases for other kinds of symmetries - if we define also new kinds of transpositions (swapping symmetric entries and leaving unchanged entries in symmetry axis/center): for example persymmetric transposition (denote it $T_p$) or centrosymmetric transposition (denote it $T_c$) and in a similar way  a skew persymmetric matrix and a skew centrosymmetric matrix  can be defined  ($ \ $ $A^{T_p}=-A$,$ \ \ $ $A^{T_c}=-A$ $ \ $ ).    


*

*I'm  interested whether there are common theorems which are fully or 
partially independent of the kind of symmetry prevailing in a matrix,
for example it seems that the inverse   of symmetric matrix (take example of $3 \times 3$ matrix) is symmetric and inverse  of persymmetric/centrosymmetric is also persymmetric/centrosymmetric matrix so the kind of symmetry is somehow preserved, why? -  maybe this depends on properties of transposition $T_x$ used for particular kind of symmetry ?   


Examples:
Input centrosymmetric matrix: 
$C=\begin{bmatrix}
5  & 4  &  1  \\
2   & \color{red}{3}   & 2  \\
1   & 4  & 5 \\
\end{bmatrix}$
Matrix Inverse:
$C^{-1}=\begin{bmatrix}
 \color{cyan}{0.875}  & \color{blue}{-2.000}  & \color{magenta}{0.625}  \\
\color{green}{-1.000}   & \color{red}{3.000}  & \color{green}{-1.000}  \\
 \color{magenta}{0.625}  & \color{blue}{-2.000}   & \color{cyan}{0.875}  \\
\end{bmatrix}$
Input persymmetric matrix:
$P=\begin{bmatrix}
 5    &  2   & \color{red}{10}  \\
 4    &  \color{red}3    &  2  \\
 \color{red}1    & 4    &  5  \\
\end{bmatrix}$
Matrix Inverse:
$P^{-1}=\begin{bmatrix}
 \color{blue}{0.054}    & \color{green}{0.233}   & \color{red}{-0.202} \\
\color{magenta}{-0.140}   &  \color{red}{0.116}   &  \color{green}{0.233} \\
 \color{red}{0.101}   & -\color{magenta}{0.140}  &   \color{blue}{0.054} \\
 \end{bmatrix}$

Sorry for brevity, however in order to give a more uniform point of view, I'd like to point out that seems to me that all the mentioned cases (and others on which I'm interested since yesterday) are of the form $A\mapsto T^tAT$ where $T\in\mathbb{H}^{n\times n}$ with $TT^*=I_n$.
