I am trying to establish an intuitive sense of what the outer product does.
I came across the following assertion on Quora:
The outer product between two vectors, U, and V, gives us a matrix. Every matrix acts as a transformation on vectors. That is, given a vector, W, their outer product transforms W into another vector. Now, this might be obvious to you, but what obviously isn’t so obvious is what it transforms it into. Specifically, $$(UV^T)W = (V\cdot W)U$$
As far as I can tell, this is wrong.
For example, given:
U = [1, 3]
V = [3, 4]
W = [2, 3]
LHS: $$UV^T=15$$ $$15*W=[30,45]$$
RHS: $$V\cdot W = 18$$ $$18*U = [18,54]$$
Based on this working, the LHS is not equivalent to the RHS.
Fwiw, if I interpret the Quora poster's vector W as the outerproduct of U and V, the assertion still does not hold up.
$$U\otimes V = W = \begin{bmatrix} 6 & 9 \\ 8 & 12 \end{bmatrix}$$
LHS: $$(UV^T)W = 15*\begin{bmatrix} 6 & 9 \\ 8 & 12\end{bmatrix}$$
RHS: $$V\cdot W = [25,75] $$ $$(V\cdot W)U = [25,75][1,3] $$
Which is also not equivalent to the assertion:
$$(UV^T)W = (V\cdot W)U$$
Two questions:
1. Am I doing something wrong here? Or am I correct in saying the Quora poster is wrong?
2. What is an intuitive description of an outer product?
The claim with your choice of $U$, $V$ ,and $W$ is $$\begin{bmatrix}1 \\ 3\end{bmatrix} \begin{bmatrix}3 & 4\end{bmatrix} \begin{bmatrix}2 \\ 3\end{bmatrix} = \left(\begin{bmatrix}3 & 4\end{bmatrix} \begin{bmatrix}2 \\ 3\end{bmatrix}\right) \begin{bmatrix}1 \\ 3\end{bmatrix}.$$ The left-hand side is $$\begin{bmatrix}3 & 4 \\ 9 & 12\end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix}18 \\ 54\end{bmatrix}.$$ The right-hand side is $$18 \begin{bmatrix}1\\3\end{bmatrix}= \begin{bmatrix}18 \\ 54\end{bmatrix}.$$
The original claim is essentially just associativity ofmatrix multiplication.