Let V = C^n and A be in Mn(C).

Let

<x,y>=sum i, j, k =1 to n Akj . conjugate Aki . x(j) . conjugate y(i)

Show this is a pre inner product on V.

Do I start by letting say B = Akj . conjugate Aki

Define <x, y> = sum i, j, k =1 to n Bx . conjugate y

& find linearity of V by the linearity of B somehow?

Thanks.