A vector $(a_{11}x, a_{22}y, a_{33}z)$ can be expended as:
$$\begin{align} \begin{pmatrix} a_{11}x \\ a_{22}y\\ a_{33}z \end{pmatrix} &= \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}\odot \begin{pmatrix} 1 & 0 & 0\\ 0& 1 &0\\ 0&0 & 1 \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} \\&= \begin{pmatrix} * & * \\ * & * \\ * & * \end{pmatrix} \begin{pmatrix} * & * & *\\ * & * & * \end{pmatrix}\odot \begin{pmatrix} 1 & 0 & 0\\ 0& 1 &0\\ 0&0 & 1 \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} \end{align}$$
but how can I write the following vector as product of matrices: $$\begin{pmatrix} exp(a_{11})x \\ exp(a_{22})y\\ exp(a_{33})z \end{pmatrix}$$ the annoying exp() prevents me doing any transformation! Maybe the question is: is there a exp() function whose parameter is a vector instead of a single number?
Your original question has changed significantly. If $$A=\left( \begin{matrix} {{a}_{11}} & 0 & 0 \\ 0 & {{a}_{22}} & 0 \\ 0 & 0 & {{a}_{33}} \\ \end{matrix} \right)$$ then $$\exp (A)\left( \begin{matrix} x \\ y \\ z \\ \end{matrix} \right)=\left( \begin{matrix} \exp ({{a}_{11}}) & 0 & 0 \\ 0 & \exp ({{a}_{22}}) & 0 \\ 0 & 0 & \exp ({{a}_{33}}) \\ \end{matrix} \right)\left( \begin{matrix} x \\ y \\ z \\ \end{matrix} \right)=\left( \begin{matrix} \exp ({{a}_{11}})x \\ \exp ({{a}_{22}})y \\ \exp ({{a}_{33}})z \\ \end{matrix} \right)$$ This is quite different from your original post which required $$\left( \begin{matrix} \exp ({{a}_{11}}x) \\ \exp ({{a}_{22}}y) \\ \exp ({{a}_{33}}z) \\ \end{matrix} \right)$$