If $A(z) :=[A\_{ij}(z)] $ and $B(z) :=[B\_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A\_{ij}(z)$, and $B\_{ij}(z) $ with 


(1). $AA^{\#}=A^{\#}A$ where $A^{\#}(z)=\left(\overline{A(\bar{z})} \right)^{T}$.


(2). $BB^{\#}=B^{\#}B$ 


(3).$ AB=BA$


Is it true that 
$$B(A A^{\#}) =(AA^{\#}) B$$. 


Thanks in advance.



Commuting matrices of complex functions
Let $z\_0\in\Bbb{R}$ be arbitrary. The matrices $A\_0:=A(z\_0)$ and $B\_0:=B(z\_0)$ are *normal*; in view of $A\_0^\*=A^\#(\bar{z\_0})=A^\#(z\_0)$ and $B\_0^\*=B^\#(\bar{z\_0})=B^\#(z\_0)$ they commute with their Hermitian adjoints. They also commute with each other. So $A\_0$ and $B\_0$ could be simultaneously diagonalized by a unitary matrix. That matrix also diagonalizes $A\_0^\*$ and $B\_0^\*$. So the four matrices $A\_0=A(z\_0),B\_0=B(z\_0),A\_0^\*=A^\#(z\_0),B\_0^\*=B^\#(z\_0)$ could be simultaneously diagonalized which implies that they all commute with each other. So $B(AA^\#)=(AA^\#)B$ at any point of the real line. Because the entries are entire functions, they coincide throughout the whole complex plane. 


**Added:** As mentioned by @Guest and @MarkSapir, $A^\#(z)$ (respectively 
$B^\#(z)$) is the Hermitian conjugate of $A(\bar{z})$ (resp. $B(\bar{z})$) rather than that of $A(z)$ (resp. $B(z)$). Indeed, the Hermitian conjugates $A^\*(z):=(\overline{A(z)})^{\rm{T}}$ and $B^\*(z):=(\overline{B(z)})^{\rm{T}}$ do not vary holomorphically with $z$; they are anti-holomorphic. On the contrary, $A^\#(z)$ and $B^\#(z)$ are holomorphic because in their definitions the complex conjugation is applied both in the domain and the range. To circumvent this difficulty, one notices that $A^\#(z)$ (respectively 
$B^\#(z)$) is indeed the Hermitian conjugate of $A(z)$ (resp. $B(z)$) if $z$ is real. So by a linear algebra argument, one can establish $B(z)\left(A(z)A^\#(z)\right)=\left(A(z)A^\#(z)\right)B(z)$ for $z\in\Bbb{R}$. This persists on the rest of $\Bbb{C}$ because the entries of these matrices are entire functions (holomorphic on the whole $\Bbb{C}$).


