# Index/Einstein notation: from/to matrix form

• Verdict
In summary, the conversation was about using index/einstein notation for matrices and vectors, and the confusion it caused for the person. They were able to get clarification on the notation and how to properly multiply matrices using this notation. The correct solutions for the given equations were D = CTBA and ABCT.
Verdict
So I've just started working with the index/einstein notation for matrices and vectors the other day. I've been doing a few exercises from a booklet I have, but I am still a bit confused. I am pretty sure my confusion is rather stupid though, so I apologize in advance.

## Homework Statement

So one question that has me puzzled is writing the following in matrix form:
Dβ$\nu$ = A$\mu\nu$B$\alpha\mu$C$\alpha\beta$

The second question, I suppose is similar, and it's writing the following as matrix multiplication:
Dαβ = Aα$\mu$B$\mu\nu$C$\beta\gamma$

## The Attempt at a Solution

Alright, so my main confusion comes from the fact that, the way I've been thought is that in order to multiply matrices they need to be the same size. So an (MxN) matrix can only be multiplied with an (RxP) one, if N = R. Now in the first, and in the second one, this is (seems?) not the case, as for example a mu by nu matrix A is multiplied with an alpha by mu matrix B. I suppose that this is not actually the case though, and that I am misunderstanding. My question is, what am I doing wrong here? Could you help me in the right direction? The second equation is giving me same issue really, as I don't know how to represent those matrices in a correct way, as to me they seem to be of the wrong dimensions.

Kind regards

All the matrices you are multiplying are square matrices. The subscripts are not the dimensions of the matrices, they are the row-column addresses of a specific element in each matrix. When you use this type of notation, you are summing over repeated indices. When you write AαμBμσ, what your are really calculating the product of the two matrices A and B. Note that you are summing over the products of the row elements of A and the column elements of B.

Chet

Mhm alright yes, that makes sense. In general they are all N by N matrices I suppose, as you let your sum (which is omitted) range to N.
However, I'm afraid I am still confused.. All the examples that I saw so far have the form AαβBβγ, so the second index of the first matrix would match the first index of the second matrix. Here, that is different. How is that reflected in the answer?
Would AμνBαμ simply be Eμμ? I suppose not, but I don't really get why..

I apologize, as this is a very basic topic, I just don't see exactly how it works just yet.

Is the answer to the first question simply

No right, there has to be more to it, as this just leaves out the indices..

Last edited:
Verdict said:
Mhm alright yes, that makes sense. In general they are all N by N matrices I suppose, as you let your sum (which is omitted) range to N.
However, I'm afraid I am still confused.. All the examples that I saw so far have the form AαβBβγ, so the second index of the first matrix would match the first index of the second matrix. Here, that is different. How is that reflected in the answer?
Would AμνBαμ simply be Eμμ? I suppose not, but I don't really get why..

I apologize, as this is a very basic topic, I just don't see exactly how it works just yet.

Is the answer to the first question simply

No right, there has to be more to it, as this just leaves out the indices..

AμνBαμ would be BA.

AμνBαμ=$\sum_{\mu=1}^{\mu=N}B_{\alpha\mu}A_{\mu\nu}$

Chestermiller said:
AμνBαμ would be BA.
More specifically, it would be ##(BA)_{\alpha\nu}##.

Remember that ##A_{\mu\nu}## and ##B_{\alpha\mu}## are just numbers, so you can reorder them: ##A_{\mu\nu}B_{\alpha\mu} = B_{\alpha\mu}A_{\mu\nu}##.

Thank you both, that makes a lot more sense. Though that leaves me a little confused.. The rearrangement works with B and A, from the aforementioned equations, but the indices of C do not match up with either B, A, or their product, do they? So how does that work? Again, I apologize for something that is probably so obvious..

Your original expressions are messed up, which is probably why you're confused. The indices should match up the way you probably think they should.

As in, they are incorrect? If that is the case I am going to punch myself for spending so much time on it instead of just asking my professor tomorrow. Thanks a lot though, at least I am more at ease with the general concept now, and I can move on to the actual topics associated with it!

Well, the second one, Dαβ = AαμBμνCβγ, is. (The first one is fine — I misread it earlier.) On the lefthand side, you have ##\alpha## and ##\beta## as free indices. On the righthand side, you have ##\alpha##, ##\beta##, ##\gamma##, and ##\nu##. It probably should have read ##D_{\alpha\beta} = A_{\alpha\mu}B_{\mu\nu}C_{\beta\nu}##.

You are entirely correct, that gamma should have been a nu, strange mistake on my part.
Giving it some more thought, the solution to the first one is probably D = CTBA, and the second one would be ABCT, or is that not a legal operation?

Yes, those are correct.

## 1. What is Index/Einstein notation?

Index/Einstein notation is a mathematical notation used to express and manipulate multilinear algebraic equations in a concise and compact way. It involves the use of index symbols to represent repeated variables in a sum or product, rather than writing out each term individually.

## 2. How is Index/Einstein notation used to convert from matrix form?

To convert from matrix form to Index/Einstein notation, the matrix elements are represented by index symbols and summed over the appropriate indices. The resulting equation will be a concise representation of the original matrix equation.

## 3. Can Index/Einstein notation be used for tensors as well?

Yes, Index/Einstein notation can be used for tensors of any rank. The notation remains the same, but the indices may have different ranges depending on the rank of the tensor.

## 4. Why is Index/Einstein notation useful in scientific calculations?

Index/Einstein notation is useful because it allows for the manipulation of complex equations in a concise and compact way. It also helps to avoid errors that may occur when writing out each term individually. Additionally, it is a standardized notation used by many scientists and mathematicians, making it easier to communicate and understand equations.

## 5. Are there any drawbacks to using Index/Einstein notation?

One potential drawback of Index/Einstein notation is that it can be difficult to read and understand for those who are not familiar with it. It also requires some practice to become proficient in using it correctly. Additionally, it may not be the most efficient notation for certain types of calculations, such as those involving matrices with a large number of elements.

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