# Write trace of AB* as summation (1 Viewer)

### Users Who Are Viewing This Thread (Users: 0, Guests: 1)

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 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.

#### Fredrik

Staff Emeritus
Gold Member
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.

Last edited:

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving