Index notation interpretation for matrices

Question

I want to understand the how to interpret the matrices which are represented by index notation. Here is my matrix

$_{}+_{}_{}−_{} _{}$

All the matrices in the equation are 3x3 matrices. Is $\sigma_{ik}$ and $\sigma_{kj}$ the same or is one the transpose of the other, and how do you identify that? The same question goes for $_{}$ and $_{}$. This is an equation from hypoelasticity model. I also understand that it is not a good practice to repeat the indices in an equation. Is that the reason they used different indices? I have got to know the basic sense of index notation, and I am in the process of learning more. It would be great if anyone can shed some light on this. Thank you.

Answer 1

I don't know about the specific equation in hand but a common way to represent matrix elements is to using $\sigma_{ij}$ to mean the element of a matrix, say $\Sigma$ at the $i$th row and the $j$th column.

Then $\sigma_{ik}\omega_{kj}$ is the product of the element of $\Sigma$ matrix at the $i$th row and the $k$th column and that of $\Omega$ matrix at the $k$th row and the $j$th column. This is useful because of matrix multiplication:

$(\Sigma \times \Omega)_{ij} = \sum_{k}{\sigma_{ik}\omega_{kj}}$

There's nothing fancy here: this is just the definition of the product of two matrices.

To give you a concrete example, suppose $\Sigma = \begin{bmatrix} 1 & -2 \\ 13 & 9 \end{bmatrix}$, then $\sigma_{11}=1, \sigma_{12}=-2, \sigma_{21}=13$ and $\sigma_{22}=9$.

Answer 2

It seems that it is Einstein's sum notation. So you have actually $A = (a_{ij})_{1\le i, j \le 3}$ with $$ a_{ij} = \sigma_{ij} + \sum_{k=1}^3 \left( \sigma_{ik} w_{kj} - w_{ik} \sigma_{kj} \right), $$ or $$ A = \Sigma + \Sigma W - W \Sigma, $$ with $\Sigma = (\sigma_{ij})_{1\le i, j \le 3}$ and $W = (w_{ij})_{1\le i, j \le 3}$.