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I'm confused about writing a system of equations in Matrix form and Einstein notation Let's start with what I understand: I have the following system of n equations (let's take n=2 to make it easier to write): $$a_{11}u_1+a_{12}u_2=b_1$$ $$a_{21}u_1+a_{22}u_2=b_2$$ I can write this in matrix form as: $$ \begin{bmatrix} a_{11} & a_{12}\\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} u_1 \\\ u_2 \end{bmatrix}= \begin{bmatrix} b_1 \\\ b_2 \end{bmatrix} $$ Or using the implied summation: $a_{ji}u_i=b_j$. Now, where I get confused is that I need to multiply the left hand side by $c_kx_k$, so my system of equation is now (u and x are two input vectors, $a_{ji}$, and $c_k$ are constant values, and $b_j$ is the RHS result): $$(c_1x_1+c_2x_2)(a_{11}u_1+a_{12}u_2)=c_1x_1a_{11}u_1+c_1x_1a_{12}u_2+c_2x_2a_{11}u_1+c_2x_2a_{12}u_2=b_1$$ $$(c_1x_1+c_2x_2)(a_{21}u_1+a_{22}u_2)=c_1x_1a_{21}u_1+c_1x_1a_{22}u_2+c_2x_2a_{21}u_1+c_2x_2a_{22}u_2=b_2$$ Can I write this system of equations using Einstein notation as follows: $c_kx_ka_{ji}u_i=b_j$? And how would I write this in a Matrix notation? I have the feeling that the parameters a and c can be combined in a 2x2x2 matrix, with the first "slice" being $c_1a_{ji}$ and the second slice $c_2a_{ji}$. But then I do not see how this matrix can be multiplied by the input vectors x and u, with are both 1x2 vectors.

Thanks for the help.

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Your Einstein notation is correct. Looking at the indices, you can see that $\mathbf{x}$ and $\mathbf{c}$ are both contracted over the same index. In terms of matrix-vector notation, you can write this as $\mathbf{c}^{\text{T}}\mathbf{x}$.

Note that after performing this operation, the result is just a number, call it $k$. You can see from your Einstein notation that none of the other quantities depend on the same indices. So your modified matrix equation could really just be written as $k\mathbf{A}\mathbf{u}=\mathbf{b}$

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  • $\begingroup$ Thanks. That makes perfect sense. I was wondering if it was possible to somehow combine $c_k$ and $a_{ji}$ and obtain a matrix with three indexes (k,j,i) that does not depend on x. $\endgroup$
    – Sam
    May 20, 2020 at 1:43
  • $\begingroup$ @Sam You could form a three index quantity, but then you are no longer working with a matrix/matrix operations. A general n-index quantity is a tensor and at that point you are probably just best sticking to the Einstein notation form. $\endgroup$
    – Tyberius
    May 20, 2020 at 1:48

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