Write trace of AB* as summation

1. Mar 20, 2014

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

2. Mar 21, 2014

Erland

A diagonal element aii of a matrix product AB depends on all elements aik of row i in A and all elements bki 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 aikbki over all i and k, not just the sum of aiibii.

3. Mar 21, 2014

Fredrik

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

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.

Last edited: Mar 21, 2014