Whether the system of matrix equations is always solvable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:31:06Z http://mathoverflow.net/feeds/question/51003 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51003/whether-the-system-of-matrix-equations-is-always-solvable Whether the system of matrix equations is always solvable yaoxiao 2011-01-03T10:38:20Z 2011-01-28T20:20:54Z <p>In recent days, I learned a linear algebra problem from one of my friends. It can be stated as follows.</p> <p>Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that the following conditions (1), (2), (3) are satisfied: \begin{align*} (1) &amp;\quad AE=EA, \cr (2) &amp;\quad BG=GB, \cr (3) &amp;\quad AF-FB=ED-CG. \end{align*} The question is whether such $E,G,F$ always exist.</p> <p>Also it is obvious that we can obtain $E,G$ by (1) and (2) easily. However the hard die is to satisfy condition (3). I just know when $A$ and $B$ have different spectra, we can obtain $F$ in a unique way.</p> http://mathoverflow.net/questions/51003/whether-the-system-of-matrix-equations-is-always-solvable/51027#51027 Answer by Kate Juschenko for Whether the system of matrix equations is always solvable Kate Juschenko 2011-01-03T15:29:01Z 2011-01-28T20:20:54Z <p>Let $E=x I_n$, $G=y I_n$, then 1-2 are satisfied and the 3rd is a system of $n^2$ linear homogeneous equations with total number of variables equals to $n^2 + 2$, thus there are simultaneously non-zero solutions. of course, one can do better estimates on the dimension of the solution.</p>