Question

Problem 4: Spectral Norm. (a) Show that ||AH A || = || A||2. (b) Show that the spectral norm is unitarily invariant, namely, ||UAV|| = unitary matrices U and V. (c) Show that = || A|| for any A 0 CE max(|| A||- || B||). 0 B

Answer 1

(a) We can write ||AH A|| as:

||AH A|| = max(||AH A x|| / ||x||)

Now, let y = AH A x. Then, we have:

||AH A x|| / ||x|| = ||y|| / ||A x||

Since ||y|| = ||A x||2 (using the fact that ||y|| = ||AH A x|| and taking the inner product of both sides with itself), we can rewrite the expres