Prove the Frobenius norm ∥·∥F is unitarily invariant:
1 Let A ∈ C^m×n with A = [α1,α2,··· ,αn]. Then∥A∥²_F =∑ⁿ_i=1∥α_i∥²₂
2 ∥A∥²_F = Tr(AHA) =∑ⁿ_i=1 λi(A^HA)
3 Let A ∈ C^m×n. For any unitary matrices U ∈U^m×m and V ∈ V^n×n, we have ∥A∥_F =∥UA∥_F = ||A^H ||_F =∥AV∥_F = ∥UAV∥_F .

The Frobenius norm ||.lle is unitarily invariant: 1 Let A E Cmxn with A = [Q1, A2, , Qn). Then n ||1|| = Ž ||0:1/12 i=1 n i=1 0 * ||A||= Tr(AHA) = Ï 1;(AHA) Let A E CMX For any unitary matrices U E Umxm and V EVnxn, have || A||| = ||UA||| = || A ||| = ||AV || | = ||UAV ||E - 3 we