# Math Help - Similar but unitary equivalent

1. ## Similar but unitary equivalent

Show that the matrices

$A=\begin{bmatrix}1&3&0\\0&2&4\\0&0&3\end{bmatrix} \ \text{and} \ \begin{bmatrix}1&0&0\\0&2&5\\0&0&3\end{bmatrix}$

are similar and satisfy $\sum_{i,j}|a_{i,j}|^2=\sum_{i,j}|b_{i,j}|^2$ but are not unitary equivalent.

I am find it had to find a matrix S such that B=SAS^{-1}.

2. ## Re: Similar but unitary equivalent

Suppose you try diagonalizing both matrices? If there is a $P$ s.t. $B=PDP^{-1},$ and there exists a $Q$ s.t. $A=QDQ^{-1},$ for the same diagonal matrix $D,$ then $P^{-1}BP=D$ and $Q^{-1}AQ=D,$ so

$P^{-1}BP=Q^{-1}AQ,$ and hence $B=PQ^{-1}AQP^{-1}.$

What is your candidate for $S,$ then?