A matrix subspace $S\subset M\_n(C)$ is called "good", if there is two linear independent elements of $S$, says $E\_1,E\_2$ which are simultaneously singular valued decomposable, i.e., $E\_1=UD\_1V$ and $E\_2=UD\_2V$ with $D\_1$, $D\_2$ diagonal and $U,V$ unitary.


Now the question becomes: if all three-dimensional subspace $S\subset M\_3(C)$ is good?


This problem is try to undertand how hard could simultaneously singular valued decomposation be, and how powerful could linear combination of matrix be. $n=3$ is the simplest case.



A 3*3 matrix space problem
wlog one can assume that $||E\_1||=1$.


Let $X$ be the span of the matrix units $E\_{11},E\_{21},E\_{31}$. Then for every $2$ linearly independent operators $E\_1, E\_2$ in this space there exists a unitary matrix $U$ such that $UE\_1=E\_{11}$, $UE\_2=\lambda E\_{21}+E\_{11}$. But $E\_{11}$ and $\lambda E\_{21}+E\_{11}$ are not simultaneously singular valued decomposable. 


