I'm continuing to prepare for a Linear Algebra exam and found another problem that puzzles me.
Let $A = I+xy^*$, where $x,y \in \mathbb{C}^m (\neq 0)$.
(a) Determine a necessary and sufficient condition on $x,y$ so that $A$ admits an eigenvalue decomposition, then find such a decomposition.
(b) Determine a necessary and sufficient condition on $x,y$ so that $A$ admits a unitary diagonalization, then find such a diagonalization.
For part (a), I know that a basis of eigenvectors for $I + xy^*$ would be sufficient; I'm not sure how to relate that condition to $x,y$ yet.
For part (b), $I +xy^*$ being normal would be sufficient to ensure a unitary diagonalization; this is equivalent (I think) to $\|x\|_2^2 \,yy^* = \|y\|_2^2 \,xx^*$. I'm not sure then how to find the diagonalization, nor what the necessary conditions are in either case.