Every matrix $A\in M\_4(\mathbb{R})$
can be represented in the form of $$A=\sum\_{i=1}^{n(A)} B\_i\otimes C\_i$$ for $B\_i,C\_i\in M\_2(\mathbb{R})$.


What is the least uniform upper bound $M$ for such $n(A)$? In other words, what is the least integer $M$ such that every $A$ admit such a representation with $n(A)\leq M$?


Is this least upper bound equal to the corresponding least upper bound for all matrices $A$ which are a matrix representation of quaternions $a+bi+cj+dk$?


As another question about tensor product representation: What is a sufficient condition for a $4\times 4$ matrix $A$ to be represented in the form of $A=B\otimes C -C\otimes B$?



Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$
Because $M\_4(\mathbb R) = M\_2(\mathbb R) \otimes M\_2(\mathbb R)$ as vector spaces (and as algebras, but we won't use this), we can replace $M\_2(\mathbb R)$ by an arbitrary $4$-dimensional vector space $V$ and $M\_4(\mathbb R)$ by $V \otimes V$. We can represent elements of $V\otimes V$ conveniently as $4\times 4$ matrices, where simple tensors are rank one matrices. The question is then equivalent to asking the minimum number of rank $1$ matrices it takes to write a $4 \times 4$ matrix as a sum of rank $1$ matrices. The answer is obviously $4$.


For quaternions acting by left multiplication by quaternions in the standard basis:


$$1= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\0 & 1 \end{pmatrix} \otimes\begin{pmatrix} 1 & 0 \\0 & 1 \end{pmatrix}$$


$$i= \begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\1 & 0 \end{pmatrix} \otimes\begin{pmatrix} 1 & 0 \\0 & 1 \end{pmatrix}$$


$$j= \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \otimes\begin{pmatrix} 0 & -1 \\1 & 0 \end{pmatrix}$$


$$k= \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes\begin{pmatrix} 0 & -1 \\1 & 0 \end{pmatrix}$$


Because only two terms appear on the right side, the matrices clearly have rank at most $2$ at tensor product, and some obtain rank exactly $2$, so the answer is two.


For the last question, the answer is isomorphic to the analogous answer for writing an element of $V \otimes V$ as $v\_1 \otimes v\_2 - v\_2 \otimes v\_1$ for $V$ a four-dimensional vector space. The condition is given by a skew-symmetry condition (i.e. 10 linear conditions) plus a Pfaffian condition (a quadratic condition). More precisely the general such matrix can be written as


$$ \begin{pmatrix} 0 & a & -a & 0 \\ b & c & d & e \\ -b & -d & -c & -e \\ 0 & f & -f & 0 \\ \end{pmatrix}$$


such that $af-be+cd=0$


