PX153 - K17 - properties of diagonalization

det A = det A^{'} = λ_{1} λ_{2} \dots λ_{n}

T r A = T r A^{'} = \sum_{i} λ_{i}

$(i i i) :$ when there are $n$ independent eigenvalues (not just for hermitian matrices), $S$ formed by placing columns as eigenvectors will diagonalize $A$
- but if $A$ is not hermitian, $S^{†} \neq S^{- 1}$ , so $S^{- 1}$ needs to be found
$(i v) :$

\begin{aligned} A & \sim B \\ A^{T} & \sim B^{T} \\ A^{- 1} & \sim B^{- 1} \end{aligned}