Understand Summation Notation & Clear Confusion

[Lucid Dreamer]

I am trying to understand summation notation and there are a few inconsistencies in my head that I would like to clear up.

Suppose C is an m*n matrix and $\vec{x}$ is a 1*m row vector. Then,

$$\vec{x}C = \sum_{i} x_{i}C_{ij} = \sum_{i} C_{ij}x_{i} = \sum_{i} {C_{ji}}^Tx_{i} = C^T \vec{x}$$

This is clearly wrong but I'm not sure which operation is wrong. In terms of dimensions it doesn't make sense since $C^T$ is n*m and $\vec{x}$ is 1*m.

 

[Sourabh N]

The discrepancy arises because this way of defining a matrix and vector does not differentiate between (e.g.) a row vector and a column vector. Have a look at this for a general introduction to Einstein's notation and this for it's implementation in your case. No issues there!

 

[HallsofIvy]

Specifically, it is saying that
$$\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & c \\ b & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$$