Suppose I have two scalars $a,b \in \mathbb{R}$ defined by the inner products:
$$ a = \mathbf{a}^{T} \mathbf{t} $$ $$ b = \mathbf{b}^{T} \mathbf{t} $$
where $\mathbf{a}, \mathbf{b}, \mathbf{t} \in \mathbb{R}^{N}$. Can I represent their product as:
$$a \cdot b = (\mathbf{a}^{T} \mathbf{t})(\mathbf{b}^{T} \mathbf{t}) $$ $$\boxed{a \cdot b = (\mathbf{a}^{T} \circ \mathbf{b}^{T})\mathbf{t}} $$
Or is there some kind of other relation?
Update:
@greg Has pointed out that this is valid only when $\mathbf{t}$ is the all zero vector, or a coordinate vector. So my next question is:
What is the relation between these two quantities for a general $\mathbf{t}$?
Answer:
In general, we have:
$$ \boxed{a \cdot b = \mathbf{1}^{T} \left( \left( \mathbf{a} \otimes \mathbf{b} \right) \circ \left(\mathbf{t} \otimes \mathbf{t} \right) \right) \mathbf{1}} $$
I don't believe the relation in the box is generally valid. Here is a simple counterexample below. I hope this helps.
$$ \textbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \textbf{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \textbf{t} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} $$
$$ \textbf{a}^{T}\textbf{t}=a=2+6=8 \\ \textbf{b}^{T}\textbf{t}=b=6+12=18 \\ a \cdot b = 8(18) = 144 $$
$$ \textbf{a}^T \circ \textbf{b}^T = \begin{pmatrix} 3 & 8 \end{pmatrix} $$ $$ \left( \textbf{a}^T \circ \textbf{b}^T \right) \textbf{t} = \begin{pmatrix} 3 & 8 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} $$
$$ 6+24=30 \ne 144 $$