Sufficient condition for Unitary equivalence

Question

Let $A = (a_{ij} )$ and $B = (b_{ij} )$ be similar, and $\sum ^n _{i,j=1} \left|a_{ij}\right|^2 = \sum ^n _{i,j=1} \left|b_{ij}\right|^2 $. Then $A$ and $B$ are unitary equivalent.

I have to prove or disprove this statement. I have tried proving this statement in various ways and come to conclusion this statement must be false. But I haven't been able to find any example. The hint given is if $A$ and $B$ are unitary equivalent then $(A+A^*)$ and $(B+B^*)$ are also.

So first can someone tell that if this statement is true or not? Then if it is true then how to prove it otherwise how to find example to disprove it?

Answer 1

The answer is no. Take $A=\begin{pmatrix}60& 90& 88\\-35& 80& -82\\21& 19& -70\end{pmatrix}$ and $B=A^T$. ($A$ is random).

Recall that when $A,B$ are real matrices, they are unitarily similar iff they are orthogonally similar.

$A,B$ are clearly similar over $\mathbb{C}$ and satisfy the relation above $tr(AA^T)=tr(A^TA)$. Yet, the only solution of the system in $X$

$AX=XB,A^TX=XB^T$ is $0$. Therefore, $A,B$ are not orthogonally similar.

Answer 2

This already fails over $\mathbb C$.

I suppose that unitary equivalence is what @user1551 means in their comment. As unitary similarity $\Rightarrow$ unitary equivalence, demonstrating a counterexample for unitary equivalence will suffice.

Consider $A=\begin{bmatrix}1&1&0\\0&0&0\\0&0&-1\end{bmatrix}$ and $B=\begin{bmatrix}1&\frac{1}{\sqrt{2}}&0\\0&0&\frac{1}{\sqrt{2}}\\0&0&-1\end{bmatrix}$. They are similar matrices with identical Frobenius norms. However, as $$A=\begin{bmatrix}1\\0\\0\end{bmatrix}\begin{bmatrix}1&1&0\end{bmatrix} +\begin{bmatrix}0\\0\\1\end{bmatrix}\begin{bmatrix}0&0&-1\end{bmatrix}, $$ $$B=\begin{bmatrix}1\\0\\0\end{bmatrix}\begin{bmatrix}1&\frac{1}{\sqrt{2}}&0\end{bmatrix} +\begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\-1\end{bmatrix}\begin{bmatrix}0&0&1\end{bmatrix}, $$ the singular values of $A$ are $\sqrt{2},1,0$ and the singular values of $B$ are $\sqrt{\frac{3}{2}},\sqrt{\frac{3}{2}},0$. Since $A$ and $B$ have different singular values, they are not unitarily equivalent and not unitarily similar to each other.