Timeline for roots of polynomial with matrix coefficients

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Aug 21, 2011 at 8:32	comment	added	spk		Thank you for your answer! But this approach becomes very hard if $m >> 1$ and $l >> 1$. So I understood, that common technique does not exist.
Aug 21, 2011 at 6:28	comment	added	Robert Israel		What I mean is something like this. Consider e.g. the case $m=2$, $l=2$ with $B_0=\pmatrix{0 & 1\cr 0 & 0\cr}$, $B_1=\pmatrix{0& 1\cr 1 & 0\cr}$, $B_2=\pmatrix{1 & 0\cr 0 & 1\cr}$. If $X= \pmatrix{x_{11} & x_{21}\cr x_{12} & x_{22}\cr}$, the system of equations is $$\begin {array}{c} x_{{2,1}}+{x_{{1,1}}}^{2}+x_{{1,2}}x_{{2,1}}=0\\ x_{{1,1}}+x_{{2,1}}x_{{1,1}}+x_{{2,2}}x_{{2,1}} =0\\ 1+x_{{2,2}}+x_{{1,1}}x_{{1,2}}+x_{{1,2}}x_{{2,2 }}=0\\ x_{{1,2}}+x_{{1,2}}x_{{2,1}}+{x_{{2,2}}}^{2}=0 \end {array}$$ You could use Groebner basis techniques to solve this system.
Aug 20, 2011 at 17:54	comment	added	spk		Thank you for the answer! As for me it is a known fact that for any field $\mathbf{F}_q$ $M_{l \times l}(\mathbf{F}_q)[x] \sim (\mathbf{F}_q[x])^{l \times l}$. But how it could help me?
Aug 20, 2011 at 1:48	history	answered	Robert Israel	CC BY-SA 3.0