As ${\bf u}, {\bf v} \in \Bbb C^n $. Under what conditions is matrix ${\bf A} = {\bf u} {\bf v}^*$ unitarily diagonalizable?
I am thinking unitarily diagonalizable also means normal. So ${\bf A}^* {\bf A} = {\bf A} {\bf A}^*$. Am I on the right track?
Just what to follow up what I have here. ${\bf A} {\bf A}^* = {\bf v} {\bf u}^* {\bf u} {\bf v}^*$ and $ {\bf A}^*{\bf A} = {\bf u} {\bf v}^* {\bf v} {\bf u}^* $
As ${\bf u}, {\bf v} \in \Bbb C^n $, ${\bf u}^* {\bf u}$ and ${\bf v}^* {\bf v}$ should be a scalar. Therefore, ${\bf A} {\bf A}^* = a {\bf v}{\bf v}^*$ and $ {\bf A}^*{\bf A} = b {\bf u} {\bf u}^* $
As ${\bf A}^* {\bf A} = {\bf A} {\bf A}^*$, $a {\bf v}{\bf v}^* = b {\bf u} {\bf u}^*$. So $ {\bf u}$ and ${\bf v}$ should be linear dependent. Please tell me if I got something wrong.