Vector notation for sum over elementwise product of 3 vectors

Question

If I have an expression for two vectors $A$ and $B$ as below:

$$\displaystyle \sum_{i=1}^N A_i B_i $$

we can write this as $ A^T B $ or $B^T A$

Now, if I have 3 vectors $A$, $B$ and $C$,

$$\displaystyle \sum_{i=1}^N A_i B_i C_i$$ how can I show that in vector notation?

Answer 1

You can write the above expression in the following way

$$\left(\begin{array}{lll}A_1\\A_2\\ \vdots \\A_N\end{array}\right)^T\left(\begin{array}{llll}B_1\\ & B_2\\ && \ddots\\ &&&B_N\end{array}\right)\left(\begin{array}{lll}C_1\\C_2\\ \vdots\\C_N\end{array}\right)$$

The Matrix in the middle is diagonal.