Due on Friday, Feb 22.
Show that every rank 1 matrix in \(M_n\) has the form \(xy^*\) for some \(x,y \in \mathbb{C}^n\).
Suppose that \(A \in M_n\) and \(\lambda \ne \mu\) are two eigenvalues of \(A\). Prove that the right eigenspace corresponding to \(\lambda\) is orthogonal to the left eigenspace corresponding to \(\mu\). Recall that two subspaces are orthogonal if every vector in one is orthogonal to every vector in the other.