Show that Hilbert-Schmidt inner product is an inner product

Question

I have to show that the Hilbert-Schmidt inner product is an inner product for complex and hermitian $d\times d$ Matrices $$(A,B)=Tr(A^\dagger B)$$ I checked the wolfram page for the definition of an inner product The first two and the last property are relatively easy to show, but I am stuck with the third property: $$(A,B)=(B,A)^*$$ For hermitian matrices I tried to show it in the following way: $$(A,B)^*=(Tr(A^\dagger B))^*=Tr((A^\dagger B)^*)=Tr((A^\dagger)^* B^*))$$

Answer 1

Maybe use the fact that $$\operatorname{Tr}(A)=\operatorname{Tr}(A^T)$$ This would lead, from your last step, to

$$\operatorname{Tr}((A^\dagger)^*B^*) = \operatorname{Tr}(A^TB^*) = \operatorname{Tr}((A^TB^*)^T) = \operatorname{Tr}((B^*)^TA) = \operatorname{Tr}(B^\dagger A) = (B,A)$$

which is the result you were searching for.