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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.

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A matrix subspace $S\in S\subset M_n(C)$ 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 $S\in S\subset M_3(C)$ 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.

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# A 3*3 matrix space problem

A matrix subspace $S\in M_n(C)$ 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 $S\in M_3(C)$ is good?