# Decomposition of a diagonal matrix into a product of particular matrices.

Could you tell me please, if it's possible to find a decomposition of a diagonal matrix $3 \times 3$ : $D ( \lambda_1 , \lambda_2 , \lambda_3 ) = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{pmatrix}$ into a product of particular matrices of type $M_i = \begin{pmatrix} 0 & a_i & 0 \\ 0 & 0 & b_i \\ c_i & 0 & 0 \end{pmatrix}$ such that : $D( \lambda_1 , \lambda_2 , \lambda_3 ) = M_1 M_2 M_3$ ?. If so, is this decomposition unique ?

We have $$M_1M_2M_3=\begin{pmatrix} a_1b_2c_3 & 0 & 0 \\ 0 & a_3b_1c_2 & 0 \\ 0 & 0 & a_2b_3c_1 \end{pmatrix}.$$ This shows that there is alway such a decomposition, and that it is not unique.
$\lambda1=a_1b_2c_3 \\ \lambda2=a_3b_1c_2 \\ \lambda3=a_2b_3c_1$
obviously the decomposition always exist and is not unique. As we can simplely assume either two of the three set $a, b, c$ is all 1 and the rest one equals $\lambda$. For instance, $a, b$ are 1 and $c$ euqals to $\lambda$.