This problem is just an exercise in notation usage. Without TeX it is next to impossible to display. However, here is a bit of guidance.

This idea is to show that this inner product is equivalent to the usual inner product defined on the space nxn matrices. I will give a good referenceLinear Algebraby Larry Smith. He shows how to work this out for <A,B>=tr{AB^T}. But his work is easily modified to this problem.