Given the $2n \times 2n $ matrix
$$A=\begin{pmatrix} \bar{A} & A^* \\ A^* & \bar{A} \end{pmatrix}$$
let $X$ the matrix such that $XX=A$, that is $X=\sqrt{A}.$
So $$X=\begin{pmatrix} X_{1,1}&X_{1,2}\\ X_{2,1}&X_{2,2} \end{pmatrix}$$ How can i show that $X_{1,2} X_{1,1} = X_{1,1}X_{1,2}$, that is that the matrices commutes?
$\def\m#1{\left[\begin{array}{c}#1\end{array}\right]}$Multiplying two block-wise centrosymmetric matrices yields $$\eqalign{ \m{A&B\\B&A}\cdot\m{X&Y\\Y&X} &= \m{(AX+BY)&(AY+BX)\\(BX+AY)&(BY+AX)} \\ }$$ Since matrix addition commutes $\big({\rm i.e.}\;(BX+AY) = (AY+BX)\big)$ the product is also block-wise centrosymmetric. Therefore any integer power of such a matrix maintains the symmetry. And any function of such a matrix (assuming it can written as a convergent power series) exhibits the same symmetry.