Let V be the set of all complex square matrices. For define by .
Using the inner product , find the adjoint of .
I feel I'm missing something because I can't arrive at an answer.
The adjoint of satisfies the condition for all B, where the angled brackets denote the inner product. If you write this in terms of the trace then it becomes . But the trace has the property that . This means that . From that you should be able to see what is.