Math 490 - Spring 2019

Homework 6

Due on Monday, March 4.

  1. If \(U \in M_n\) is unitary and \(\lambda \in \sigma(U)\), then show that \(| \lambda | = 1\).

  2. If \(U \in M_n\) is unitary, then show that \(|\det U| = 1\).

  3. Describe all possible \(n\)-by-\(n\) diagonal unitary matrices with real entries.

  4. 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.

  5. Prove that unitary matrices preserve inner products. That is, for any \(U \in M_n\) and \(x,y \in \mathbb{C}^n\), \(\langle Ux,Uy \rangle = \langle x, y \rangle\).

  6. 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.)