Answer to Question #24014 in Linear Algebra for Sabrina DelCourt

Question

a) Show that the complex dot product vec(A).vec(B) = vec(B)^(conjugate transpose)[vec(A)] can be obtained by:
vec(A).vec(B) = trace(AB^(conjugate transpose)) = tr(AB^(conjugate transpose)).
We can therefore use the trace to define an inner product between matrices: <A,B>=
trace (AB^(conjugate transpose)).

b) Show that trace(AA^(conjugate transpose)) >= 0 for all A, so that we can use the trace to define a norm on matrices: ||A||^2_F = trace (AA^(conjugate transpose)). This norm is the Frobenius norm.