I have to solve an equation that involves a matrix, two vectors and logarithm inbetween. $$\log(s_{2}) = M \log(s_{1})$$ $M$ is a $n\times n$ matrix, $s_{1}$ and $s_{2}$ are two $n \times 1$ vectors. I want an expressions linking s2 to s1 without a log between them, so I wanted to use the $\exp()$ but as the right side of the equation is a product of matrix and vector I am not sure whether this is allowed or not (I highly doubt it is) , and what would be the resulting relation.
Is it possible ?
Your equation is really a set of equations $$\begin{bmatrix}\log((s_2)_1)\\\log((s_2)_2)\\\vdots\\\log((s_2)_n)\end{bmatrix}=M\cdot \begin{bmatrix}\log((s_1)_1)\\\log((s_1)_2\\\vdots\\\log((s_1)_n\end{bmatrix}$$
So, if you introduce a new variable $x_i = \log((s_2)_i)$ and $y_i=\log((s_1)_i)$, you can solve the matrix equation $$x = M\cdot y$$ (where $x=[x_1,x_2\dots, x_n]^T$, of course).
Then, once you have $x$, you have $x_i$ for each $i$ and you can calculate $(s_2)_i=e^{x_i}$