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If $A$ is an $n\times n$ invertible matrix of complex functions entries, with the property $AA^{\#}=A^{\#}A$ where $A^{\#}(z)=\left(\overline{A(\bar{z})} \right)^{T}$. Is it true that $$A B A^{\#}=A^{\#}BA$$ for any $n\times n$ complex matrix of functions $B$?

If not, when this result could be true? Thanks in advance.

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E.g. try $$ A = \pmatrix{z+i & z\cr z & z+i\cr},\ A^\# = \pmatrix{z-i & z\cr z & z-i\cr}, \ B = \pmatrix{1 & 0\cr 0 & 0\cr} $$

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