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 ?


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:

$P^{-1}=\begin{bmatrix} \color{blue}{0.054} & \color{green}{0.233} & \color{red}{-0.202} \\ \color{magenta}{-0.140} & \color{red}{0.116} & \color{green}{0.233} \\ \color{red}{0.101} & -\color{magenta}{0.140} & \color{blue}{0.054} \\ \end{bmatrix}$

  • 1
    $\begingroup$ Not to elaborate a full answer, but the intuition you adquire in elementary linear algebra is to think all the way of autovalors and autovectors. Symmetry (and lack of it) about main diagonal implies autovectors a lot, so it's the first point you wonder when faced to a matrix. In quantum mechanics this symmetry is the definition for the class of relevant operators-matrices $\endgroup$ – Rafa Budría Mar 11 '17 at 10:52
  • $\begingroup$ @RafaBudría I'm sorry I have a very little knowledge of quantum mechanics so it tells me not too much.. $\endgroup$ – Widawensen Mar 11 '17 at 10:54
  • $\begingroup$ In QM, operators (matrices in finite dimensional problems) have to be hermitians: equal to its own conjugate transpose $\endgroup$ – Rafa Budría Mar 11 '17 at 10:58
  • $\begingroup$ @RafaBudría Interesting, this is a reason from QM. $\endgroup$ – Widawensen Mar 11 '17 at 11:00

Let me start by giving my interpretation of what I think you mean by symmetry in this context. I’ll then provide some information that fits within this interpretation, which you may find useful.

Based on your examples of symmetry, persymmetry, and centrosymmetry, I think that you’re focusing on linear involutory actions $f$ on square matrices over a field (I'll assume the real or complex numbers) and the class of matrices that are invariant under such actions. Linear, of course, means that $f(cA+dB)=cf(A) + df(B)$ for equal sized square matrices $A$ and $B$ and scalars $c$ and $d.$ By an involutory action $f$ on a square matrix, I mean actions that, when performed twice in succession ($f^2$), leave the matrix unchanged (effectively a $\mathbb{Z_2}$ type of action). Examples of such linear involutory actions include transposition (symmetric matrices being the matrices left unchanged under this operation) and simultaneous pre-and post-multiplication by the exchange matrix $J$ (centrosymmetric matrices being the matrices left unchanged under this operation). You might further be restricting your attention to cases where matrix elements are merely permuted in some involutory fashion (giving a visually evident symmetry), but I won’t assume that here.

So, let’s denote the linear involutory action $f$ on a square matrix $A$ by $f(A)$, where again by involutory we mean $f^2(A)=A.$ For such actions, we can always decompose the matrix $A$ into an “$f$-symmetric” part $A^+$ and an “$f$-skew-symmetric” part $A^-$ by setting $A^+=\frac{1}{2}(A + f(A))$ and $A^-=\frac{1}{2}(A - f(A))$. With these definitions, it’s clear that $A=A^+ + A^-,$ $ A^+ = f(A^+),$ and $ A^- = -f(A^-).$

The following is a partial list of some familiar involutory actions $f$ that can be applied to a square matrix and that also satisfy linearity:

  1. Negation
  2. Complex conjugation
  3. Transposition
  4. Simultaneous pre and post multiplication by the exchange (backward-identity) matrix $J.$

Other linear involutory actions can be constructed by combining these actions. The square matrices which are invariant under some of these combined actions lead to the classes of matrices you mentioned. Others are less interesting, such as those invariant under only item 1 (just the zero matrix) and those invariant under only item 2 (the real matrices).

For example, persymmetric matrices are those matrices invariant simultaneously under items 3 and 4: that is, matrices satisfying $A=JA^TJ.$ Hermitian matrices are those matrices invariant simultaneously under items 2 and 3: $A=\bar{A^T}$. Centrosymmetric matrices are those matrices invariant under item 4: that is, matrices satisfying $A=JAJ.$ And there is also some literature about Centro-Hermitian matrices which combine items 2 and 4: $A=J\bar{A}J.$ Of course, the skew-counterparts to all of these matrix classes can be obtained by additionally throwing in item 1.

Regarding matrix inversion for elements of $GL_n(F),$ note that inversion is also an involutory action (albeit non-linear) and behaves well on the classes of matrices invariant under the four items on the list individually and in combination. For item 3, the key result is the basic fact that if $A=A^T$ then $A^{-1}=A^{-T}.$ You can leverage off of this fact to establish that the inverse of invertible elements of other classes of matrices you mentioned remain within those classes.

As one prototypical example, a skew-persymmetric matrix $A$ by definition satisfies $A=-JA^TJ.$ Then, since $J = J^{-1},$ we immediately have that $A^{-1}=-J(A^{-1})^TJ$ which shows that $A^{-1}$ is also skew-persymmetric.


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