Show that the complex dot product veca vecb vecbveca can

Question

Question: (a) Show that the complex dot product vec(A)· vec(B) = vec(B)^†vec(A) can be obtained by

vec(A). vec(B) = trace(AB^†) = tr(AB^†),

Where, for a square matrix C, trace (C) means the sum of the entries along the main diagonal of C. We can therefore use the trace to define an inner product between matrices: < A, B >= trace (AB†).

(b) Show that trace (AA^†) ≥ 0 for all A, so that we can use the trace to define a norm on matrices: ||A||² = trace (AA^†).