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

Question

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 ?

Thank you in advance for your help.

Answer 1

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.

Answer 2

If the decomposition exists, the below equations shall hold:

$\lambda1=a_1b_2c_3 \\ \lambda2=a_3b_1c_2 \\ \lambda3=a_2b_3c_1$

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 $\lambda$.