Let $A,B\in M_n(F)$. Set $C(A)=\{X\in M_n(F) \mid XA=AX\}$. Similarly $C(B)$ is defined. If $C(A)\subseteq C(B)$, I have to show $B\in F[A]$. ($F$ is a field.)
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What's definition of $F[A]$ ? – Mojtaba Jun 07 '15 at 18:03
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1http://math.stackexchange.com/questions/497806/matrices-b-that-commute-with-every-matrix-c-commuting-with-a – user26857 Jun 07 '15 at 18:22
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This is a well-known theorem contained in Wedderburn's "Lectures on matrices", page 106.
Theorem (Wedderburn): If the matrix $B$ commutes with every matrix that commutes with $A$ then $B$ is a scalar polynomial of $A$.
See here for a proof, or this article.
Dietrich Burde
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