Decomposing a complex square matrix into multiple square matrices

Question

I am working on a problem in optics (called Multi-plane light conversion) where I am trying to find the decomposition of a target transformation (matrix) $\hat{T}\in\mathbb{C}^{(NxN)}$. The form of decomposition I am looking for is something like: $$\hat{T}=\hat{u_n}...\hat{u_p}\hat{u_2}\hat{u_p}\hat{u_1}$$ where $\hat{u_k}\in\mathbb{C}$, where $k=1,2,...,n$, are the matrices (they can be diagonal) that I am trying to find and $\hat{u_p}$ is a known matrix (not necessarily diagonal). So for a given $n$, let's say $n=3$, the target matrix is: $$ \hat{T}=\hat{u_3}\hat{u_p}\hat{u_2}\hat{u_p}\hat{u_1}$$ I really would appreciate your help, any useful resource or advice is welcomed! Thank you so much in advance!