0
$\begingroup$

In recent days, I learned a linear algebra problem from one of my friends. It can be stated as follows.

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: $$ \begin{align*} (1) &\quad AE=EA, \cr (2) &\quad BG=GB, \cr (3) &\quad AF-FB=ED-CG. \end{align*} $$ The question is whether such $E,G,F$ always exist.

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.

$\endgroup$
7
  • 5
    $\begingroup$ A solution is E=G=F=0. $\endgroup$
    – Did
    Jan 3 '11 at 10:43
  • 1
    $\begingroup$ $E$ is usually the identity... $\endgroup$ Jan 3 '11 at 11:37
  • 2
    $\begingroup$ @Wadim: ...then why (1)? Anyway, my point was that the OP could wish to state the question more precisely. And a bit of context would not hurt either. $\endgroup$
    – Did
    Jan 3 '11 at 13:13
  • 2
    $\begingroup$ @yaoxiao: it is considered impolite to edit your question in a way that makes existing answers unintelligible. I have restored the question. $\endgroup$ Jan 6 '11 at 16:44
  • 1
    $\begingroup$ I have flagged for the moderators to lock the question. @yaoxiao: What you are doing now is spam. $\endgroup$ Jan 6 '11 at 16:53
9
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.