Due on Monday, March 4.
If \(U \in M_n\) is unitary and \(\lambda \in \sigma(U)\), then show that \(| \lambda | = 1\).
If \(U \in M_n\) is unitary, then show that \(|\det U| = 1\).
Describe all possible \(n\)-by-\(n\) diagonal unitary matrices with real entries.
For a unitary matrix \(U \in M_n\) with eigenvalue \(\lambda\), prove that all right eigenvectors of \(U\) corresponding to \(\lambda\) are also left eigenvectors.
Prove that unitary matrices preserve inner products. That is, for any \(U \in M_n\) and \(x,y \in \mathbb{C}^n\), \(\langle Ux,Uy \rangle = \langle x, y \rangle\).
Prove that the product \(UV\) of two unitary matrices \(U, V \in M_n\) is also unitary. (In fact, the set of \(n\)-by-\(n\) unitary matrices is a group under matrix multiplication.)