Is there a way to break down or approximate an $m \times n$ matrix into a product of $m \times k$, $k \times k$ and $k \times n$ matrices?

Question

Is there a way to break down or approximate an $m \times n$ matrix into a product of $m \times k$, $k \times k$ and $k \times n$ matrices, with a certain amount of error? I am not looking for SVD because here, unlike in SVD, $k> m,n$. Also, m,k,n $\neq 1$. For example, say a $4 \times 3$ matrix into $4\times 8$, $8 \times 8$ and a $8 \times 3$ matrix. I have been scouring the internet to see if there exists a matrix decomposition of sorts to do this. I would like to know if there is any such in existence.

I am basically trying to solve for X in matrix equation of the format : $AXB=C$ where X is a diagonal matrix. Matrices $A,B$ and $C$ are known. They can be square or rectangular, complex valued matrices.

Thanks.

Answer 1

Consider rank of the matrices on the right - it is not bigger than min of rank of kxk matrix and rank of kxn one. Hence it is not bigger than k. On the left side it can be min of {n,m}.

Answer 2

As various comments said, with $k>m,n$ there are many solutions, including the simple:

$$\begin{bmatrix}a &b & c \\d&e&f\\g&h&i\\j&k&l\end{bmatrix} = \begin{bmatrix}a &b & c &0&0&0&0&0\\d&e&f &0&0&0&0&0\\g&h&i &0&0&0&0&0\\j&k&l &0&0&0&0&0\end{bmatrix} \begin{bmatrix}1 &0 & 0 &0&0&0&0&0 \\0 &1 &0 & 0 &0&0&0&0\\0 &0 &1 &0 & 0 &0&0&0\\0&0 &0 &1 &0 & 0 &0&0\\0&0&0 &0 &1 &0 & 0 &0\\0&0&0&0 &0 &1 & 0 &0\\0&0&0&0&0 &0 &1 &0 \\ 0 &0&0&0&0&0 &0 &1\end{bmatrix} \begin{bmatrix}1 &0 & 0 \\ 0 &1 &0 \\0 &0 &1 \\0&0 &0 \\0&0 &0 \\0&0 &0 \\0&0 &0 \\ 0&0 &0 \end{bmatrix}$$