Problem with notation of matrix elements

[Lambda96]

TL;DR

What does this notation $A_{\quad i}^j$ and $A_i^{\quad j}$ mean?

Hi,

In one of my assignments, we had to prove that the Frobenius product corresponds to a complex scalar product. For one, we had to prove that the Frobenius product is hermitian symmetric.

I have now received the solution to the problem, and unfortunately I do not understand the notation for the individual matrix elements. I only know the notation $a_{ij}$ but what does it mean when one of the indices is written with a space of A or B, what is this space about? What should be the row and what the column in this kind of notation?

Here is the solution

 

[fresh_42]

TL;DR Summary: What does this notation $A_{\quad i}^j$ and $A_i^{\quad j}$ mean?

Hi,

In one of my assignments, we had to prove that the Frobenius product corresponds to a complex scalar product. For one, we had to prove that the Frobenius product is hermitian symmetric.

I have now received the solution to the problem, and unfortunately I do not understand the notation for the individual matrix elements. I only know the notation $a_{ij}$ but what does it mean when one of the indices is written with a space of A or B, what is this space about? What should be the row and what the column in this kind of notation?

Here is the solution

View attachment 329456

It is called Einstein notation or Einstein summation. Physicists use it all the time.
https://en.wikipedia.org/wiki/Einstein_notation

You can deconstruct it by the image you posted.
\begin{align*}
(A^\dagger B)_{ij}&=\sum_{k=1}^n (A^\dagger )_{ik}\cdot B_{kj} =\sum_{k=1}^n (\overline{A_{ki}})\cdot B_{kj}\\
\operatorname{trace}(A^\dagger B)&=\sum_{p=1}^n (A^\dagger B)_{pp}\\
&=\sum_{p=1}^n \left(\sum_{k=1}^n (\overline{A})_{kp}\cdot B_{kp}\right)\\
&=\sum_{j=1}^n \left(\sum_{i=1}^n (\overline{A})_{ij}\cdot B_{ij}\right)\\
&= (\overline{{A_j}^i})\cdot {B^j}_i
\end{align*}

It is an abbreviation for the summation. Summed is over the indices that occur on top and at the bottom, here twice: sum over $i$ and sum over $j$.

 

[PeroK]

TL;DR Summary: What does this notation $A_{\quad i}^j$ and $A_i^{\quad j}$ mean?

Hi,

In one of my assignments, we had to prove that the Frobenius product corresponds to a complex scalar product. For one, we had to prove that the Frobenius product is hermitian symmetric.

I have now received the solution to the problem, and unfortunately I do not understand the notation for the individual matrix elements. I only know the notation $a_{ij}$ but what does it mean when one of the indices is written with a space of A or B, what is this space about? What should be the row and what the column in this kind of notation?

Here is the solution

View attachment 329456

If in an assignment you encounter a notation that you have never seen before, then there must be a serious disconnection between your course syllabus and what you are studying.

 

[Lambda96]

Thanks fresh_42 for your help