Hermitian Matrices $AA^{*}=CC^{*}$,

Question

Could any one help me proving or disproving this statement:

If $A,B,C$ are $n\times n$ invertible matrices, with $AA^{*}=CC^{*}$, and $B^{*}=B$, does it follow that $$ ABA^{*}=CBC^{*} $$

thanks in advance!

Answer 1

Here is one counter example with $2\times 2$ matrices.

$A=\begin{pmatrix} 1 & i \\ i & 1\end{pmatrix}\quad A^*=\begin{pmatrix} 1 & -i \\ -i & 1\end{pmatrix}\quad AA^*=\begin{pmatrix} 2 & 0 \\ 0 & 2\end{pmatrix}$

$C=\begin{pmatrix} 0 & i\sqrt{2} \\ i\sqrt{2} & 0\end{pmatrix}\quad C^*=\begin{pmatrix} 0 & -i\sqrt{2} \\ -i\sqrt{2} & 0\end{pmatrix}\quad CC^*=\begin{pmatrix} 2 & 0 \\ 0 & 2\end{pmatrix}$

$B=\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}$

Then we have

$ABA^*=\begin{pmatrix} 0 & 2i \\ -2i & 0\end{pmatrix}\quad CBC^*=\begin{pmatrix} 2 & 0 \\ 0 & -2\end{pmatrix}$