How Do You Write the Trace of AB* as a Summation?

[catsarebad]

i'm kinda confused regarding summation so I'm hoping someone can help me figure this out and explain to me why it is the way it is

trace(AB*) = ? in summation form

* = adjoint = conjugate and transpose = transpose and conjugate

assume both matrices are square mx of same size n x n

trace = sum of diagonal entries

i'm got this after brute force

(summation of this entire thing) a_ij x conjugate of (b_ij)

i, j runs from 1 to n


but somehow I'm thinking it should be

(summation of this entire thing) a_ii x conjugate of (b_ii)

i runs from 1 to n

 

[Erland]

A diagonal element a_ii of a matrix product AB depends on all elements a_ik of row i in A and all elements b_ki of column i in B, not just on the diagonal elements of A and B. Therefore, the trace of AB must be the sum of a_ikb_ki over all i and k, not just the sum of a_iib_ii.

 

[Fredrik]

It sounds like what you're confused about isn't summation, but rather the definition of matrix multiplication. I'll quote myself:

I will denote the entry on row i, column j of an arbitrary matrix X by $X_{ij}$. The definition of matrix multiplication says that if A is an m×n matrix, and B is an n×p matrix, then AB is the m×p matrix such that for all $i\in\{1,\dots,m\}$ and all $j\in\{1,\dots,p\}$,
$$(AB)_{ij}=\sum_{k=1}^n A_{ik}B_{kj}.$$

The problem is quite easy if you just use the definitions. The other definitions you need to use are $\operatorname{Tr X}=\sum_{i}X_{ii}$ and $(X^*)_{ij}=(X_{ji})^*$, where the first * denotes the adjoint operation and the second one denotes complex conjugation. But you don't seem to be confused about those.