At the moment, I am working with Runge-Kutta methods. We can assume that this is given by a Butcher tableau, where $A$, $b$, $c$ can be extracted from.
Consider the matrix $M$, with $M_{ij} := b_i A_{ij} + A_{ji} b_j - b_i b_j$. For all $j$, we have that $b_j > 0$ and that $c \in [0,1]^s$.
In an exercise class, I need to show that
$$ M = \sum_{l=1}^s \mu_l \mu^T_l, \quad \text{$\mu_i \in \mathbb{R}^s$ for $i = 1, \ldots, s.$}$$
So, what I am looking for is a rank-$1$ decomposition of the given matrix $M$.
I tried looking at case $s=2$, but didn't come to any meaningful conclusion. It looks like this:
$$ M = \begin{pmatrix} 2b_1 a_{11} - b_1^2 & a_{21} b_2 + b_1 a_{12} - b_1 b_2\\ a_{21} b_2 + b_1 a_{12} - b_1 b_2 & 2b_2 a_{22} - b_2^2 \end{pmatrix} = \begin{pmatrix} \mu_{11}^2 + \mu_{21}^2 & 2\mu_{12} \mu_{21} \\ 2\mu_{12} \mu_{21} & \mu_{12}^2 + \mu_{22}^2 \end{pmatrix}.$$
I tried expending and making squares, but it doesn't seem to work, e.g.:
$$ a_{11} = 2b_1 a_{11} - b_1^2 - a_{11}^2 + a_{11}^2 = -(a_{11}- b_1)^2 + a_{11}^2.$$
I also thought about applying an adaptation of Cholesky decomposition, but with no success either.
Any hints? It seems that there I need to have more constraints, because for instance for $2 a_{11} < b_1$ in $a_{11}$, I will not get anything of the form form the RHS. So, are there assumptions that I am not considering?