What is a unitary matrix? What does it mean to say that a matr... | Filo

Matrix Definition & Examples

(c) Use properties of the trace to show that for any unitary Q ∈ℝ^n × n (i.e., Q^T Q = I),A - BQF^2 = tr(A^T A) tr(B^T B) - 2(tr(Q^T B^T A)).(This implies A - BQF is minimized by the unitary matrix Q that maximizes tr(Q^T B^T A).)(d) Let B^T A = U Σ V^T ∈ℝ^n × n denote the full singular value decomposition of B^T A. Show thattr(Q^T B^T A) = tr(Σ Z),where Z = V^T Q^T U.(e) Show that Z is unitary (Z^T Z = I), and explain why this implies that tr(Σ Z) is maximizedwhen Z = I. Then explain why this