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

In summary: It sounds like you're confused about the distributive law for matrices.In summary, the trace of an m×n matrix A is the sum of the diagonal elements of A.
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

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.

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

Fredrik said:
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.

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## 1. What is the purpose of writing the trace of AB* as a summation?

The purpose of writing the trace of AB* as a summation is to express the trace of the product of two matrices, AB*, as a sum of individual elements. This can be useful in simplifying calculations and understanding the properties of the matrices involved.

## 2. How do you write the trace of AB* as a summation?

To write the trace of AB* as a summation, you first need to multiply the matrices AB* and then take the trace of the resulting matrix. The trace of a matrix is the sum of its diagonal elements. Therefore, you can express the trace of AB* as a summation of the diagonal elements of the product matrix.

## 3. What are the properties of the trace of a matrix?

The trace of a matrix has several properties, including linearity, cyclic property, and invariance under similarity transformations. These properties can be useful in simplifying calculations and solving problems involving matrices.

## 4. Can the trace of AB* be written as a product instead of a summation?

No, the trace of AB* cannot be written as a product. The trace of a product of two matrices is always equal to the product of the individual traces in reverse order. Therefore, the trace of AB* cannot be expressed as a product of the traces of A and B.

## 5. How can the trace of AB* be used in real-world applications?

The trace of AB* has many applications in various fields, including physics, engineering, and computer science. It can be used to calculate the total energy of a system in physics, model fluid dynamics in engineering, and optimize algorithms in computer science, among others.

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