<s>[INST] Question about matrix multiplication notation I have the following matrices: 
$A=\begin{pmatrix}
        -\frac{2}{3} & \frac{1}{3} & 0 \\
        \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\
        \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \\
        \end{pmatrix}$
; $B=\begin{pmatrix}
        2 & 3\\
        2 & 0 \\
        0 & 3\\
        \end{pmatrix}$
; $C=\begin{pmatrix}
        6 & 3& 4 \\
        6 & 6 & 0 \\
        \end{pmatrix}$
;$u=\begin{pmatrix}2 \\ -5 \\ 4 \end{pmatrix}$
I want to calculate the following:
$\sum_j a_{ij}u_j$
$\sum_{jk} c_{ij}a_{jk}b_{kl}$
but I am not used to this kind of notation. Does the first one mean to multiply all $i$-rows of A with the $j$-column of u?
$\implies AB=\begin{pmatrix}
        -\frac{2}{3} & \frac{1}{3} & 0 \\
        \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\
        \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \\
        \end{pmatrix} \cdot \begin{pmatrix}2 \\ -5 \\ 4 \end{pmatrix}=\begin{pmatrix}-3 \\ 4 \\ 0 \end{pmatrix}$?
I am not sure what the second one is trying to say? Maybe $C\cdot A\cdot B$?
 [/INST] $a_{ij}$ denotes the $i^{th}$ row and $j^{th}$ column of $A$. The first summation means keeping $i$ fixed, increment the $j$ parameter as below for each $i$.
$$\sum_j a_{ij}u_j = a_{i1}u_1+a_{i2}u_2+a_{i3}u_3$$
This translates to the product $A.U$.
Similarly for the second sum.
</s>