Denis Serre's suggestions all sound very sensible. Here are a few which I think haven't yet been mentioned, and which might be at the right level even if they are "lower-priority" or "too specialized".
Tridiagonalization of symmetric matrices (why not apply this to the GOE, ahem)
Unitary similarity to an upper triangular matrix
However, I guess it depends on the type of linear algebra course you can assume the students have taken...