I came across a matrix of the form $\begin{bmatrix} a& b& 0& 0\\0& 0& c& d\end{bmatrix}$. At first glance, I kinda felt it can be written as a Kronecker product. I tried doing so but the best that I can do is the following: $$\begin{bmatrix}1& 0\\ 0&0\end{bmatrix}\otimes[a\ b]+\begin{bmatrix}0& 0\\0& 1\end{bmatrix}\otimes [c \ d]. $$
Is there a way to write this as a single Kronecker product?
I'm using Horn & Johnson's Kronecker product $A\otimes B=[a_{ij}B]$. I don't prescribe the size of the factors except it has to be smaller than the original to avoid $A=A\otimes[1]$ in the comments. If it can be written like so, then either the factors are
- $2\times 2$ and $1\times 2$
- $2\times 1$ and $1\times 4$.
My attempt was under case 1). Can it be improved? If not, can one do it in case 2)?
It can only be done if $(a,b)$ and $(c,d)$ are linearly dependent. Say $(c,d)=\lambda(a,b)$. Then $$\begin{pmatrix}1&0\\0&\lambda\end{pmatrix}\otimes\begin{pmatrix}a&b\end{pmatrix}.$$