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I am trying to learn some linear algebra, and currently I am having a difficulty time grasping some of the concepts. I have this problem I found that I have no idea how to start.

Assume that $\bf A$ is an $n\times n$ complex matrix which has a cyclic vector. Prove that if $\bf B$ is an $n\times n$ complex matrix that commutes with $\bf A$, then ${\bf B}=p({\bf A})$ for some polynomial $p$.

All I know at this point is that ${\bf AB}={\bf BA}$.

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    Do you know what it means for a matrix to have a cyclic vector?2012-08-04
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    @Gerry, I think you have hit it. See her other question. P.S. this is the first time I have heard of a cyclic vector.2012-08-04
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    A cyclic vector of $A$ is an element $v \in C^{n}$ such that $A^{0}v,A{1}v,...,A^{n-1}v$ are linearly independent. @ Will...I went to your link, and I saw the Thm, but I do not understand how to show that (I) implies 2-4 statements. Yes, I am new to this, but I want to learn it.2012-08-04
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    @Beth, I give some links at the other answer, and mention at least one book. Any part of the equivalence is rather long for a website answer, but I give an easy example where an eigenvalue in two separate Jordan blocks makes a specific problem.2012-08-04

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