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I am currently reading this article about companion matrices [wikipedia][1]

[1]: http://en.wikipedia.org/wiki/Companion_matrix . This brought me to the following question: I guess every companion matrix is similar to ONE Jordan block right

I think the line in wikipedia that says that minimal and characteristic polynomial coincide, explicitely "the characteristic polynomial of A coincides with the minimal polynomial of A, equivalently the minimal polynomial has degree n" is wrong, as it could differ by a sign, but it is just a guess, I do not know this.

Assume we have a cyclic subspace $U\subset V$ for an endomorphism $A$ is it true that $A|_U$ is similar to a Jordan block? (I assume this to be true, cause if a cyclic vector space has a representation similar to a companion matrix, then this just follows from my first question). But this would give me a deep inside how Jordan Blocks and cyclic vector spaces are related to each other...

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    see http://math.stackexchange.com/questions/92480/given-a-matrix-is-there-always-another-matrix-which-commutes-with-it/92832#92832 – Will Jagy Sep 10 '13 at 18:59
  • actually, although I see that this might answer my question, I only see that the char-poly and the min-poly coincide, could you just tell me whether a companion matrix is then always similar to one jordan block? –  Sep 10 '13 at 19:07
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    No. You may have more than one eigenvalue. However, for a companion matrix, there is just a single Jordan block for each eigenvalue. – Will Jagy Sep 10 '13 at 19:14

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Let $U$ be an $A$- cyclic subspace of a vector space $V$. Then the matrix representation $[A|_U]$ of $A|_U$ is a companion matrix and equivalently $U$ is an $A$-cyclic module. However, this does not mean that $U$ is an indecomposable $A$-cyclic module (this is what would give a single Jordan block, over an algebraically closed field). What will determine the Jordan structure of $[A|_U]$ is the form of the minimal polynomial $m_{A|_U}$ of $A|_U$. For example, if $m_{A|_U}$ has more than one distinct roots, then certainly we are going to have more than one Jordan blocks.

Exercise: construct a polynomial with more than one distinct roots and find a companion matrix whose characteristic polynomial is the polynomial you constructed. Then find the Jordan form of this matrix.

Manos
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