Matrix Analysis - Reading Course

Math 490 - Spring 2019

Homework 10

Due on Friday, April 5.

Recall that a matrix \(P \in M_n\) is an orthogonal projection if \(P^* = P\) and \(P^2 = P\).

1. Prove that if \(P\) is an orthogonal projection, then \(I-P\) is also an orthogonal projection.

2. Prove that all eigenvalues of an orthogonal projection \(P\) are 0 and 1.

3. Prove that a normal matrix is an orthogonal projection if and only if all of its eigevalues are \(0\) or \(1\).
   

4. Show that if \(A \in M_n\) is normal, then \(\|Ax\| = \|A^*x\|\) for all \(x \in \mathbb{C}^n\).

5. Show that a normal matrix is unitary if and only if all of its eigenvalues have absolute value 1.

6. Show that a normal matrix is Hermitian if and only if all of its eigenvalues are real.