Matrix notion for a double summation My question is some-what tied to this one:
Is there any matrix notation the summation of a vector?
Is there a way to express this double summation into an equivalent product of matrices/vectors?
$\mu=\sum_{i=1}^{N}\sum_{j=1}^{N}b_i b_jX_{i,j}\cos\left ( \sigma_i+\sigma_j \right )$
$\mu$ is a scalar. My feeling is this should be straightforward, but the solution eludes me beyond the inner summation. 
 A: We can start with a quadratic form
for vector $\mathbf v = (v_1, v_2, \ldots, v_N)$
and array $A$,
which already has some resemblance to what you want:
$$ \mathbf v^T A\, \mathbf v =\sum_{i=1}^N \sum_{j=1}^N A_{i,j} v_i v_j.$$
All that is missing is the factor of $\cos\left(\sigma_i + \sigma_j\right)$.
An obvious way to deal with that is to set
$A_{i,j} = X_{i,j} \cos\left(\sigma_i+\sigma_j\right)$
for $i = 1, \ldots, N$ and $j = 1, \ldots, N$,  in which case
$$ \sum_{i=1}^N \sum_{j=1}^N b_i b_j X_{i,j} \cos\left(\sigma_i+\sigma_j\right) = \mathbf b^T A\, \mathbf b, $$
though I suspect that's probably not very satisfying.
Alternatively, use the identity
$$ \cos\left(\sigma_i + \sigma_j\right) 
= \cos\sigma_i \cos\sigma_j - \sin\sigma_i \sin\sigma_j $$
to write
\begin{align}
\sum_{i=1}^N \sum_{j=1}^N b_i b_j X_{i,j} \cos\left(\sigma_i+\sigma_j\right)
&= \sum_{i=1}^N \sum_{j=1}^N b_i b_j X_{i,j} \cos\sigma_i \cos\sigma_j
  - \sum_{i=1}^N \sum_{j=1}^N b_i b_j X_{i,j} \sin\sigma_i \sin\sigma_j \\
&= \sum_{i=1}^N \sum_{j=1}^N X_{i,j} (b_i \cos\sigma_i) (b_j \cos\sigma_j) \\
& \qquad
  - \sum_{i=1}^N \sum_{j=1}^N X_{i,j} (b_i \sin\sigma_i)(b_j \sin\sigma_j) \\
&= \mathbf u^T X\, \mathbf u - \mathbf v^T X\, \mathbf v
\end{align}
where $u_i = b_i \cos\sigma_i$ and $v_i = b_i \sin\sigma_i$,
$i = 1, \ldots, N$.
