Problem Let A ($n\times n$ matrix) be a single Jordan block and let $C$ be an $n\times n$ matrix that commutes with $A$. Prove that $C = f(A)$ for some polynomial $f$.

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    $\begingroup$ What have you tried? If you keep asking questions without even waiting for the answer of the previous ones, we are going to end up concluding that you want us to do your homework... $\endgroup$ – Mariano Suárez-Álvarez Nov 20 '11 at 16:39

One way to do this is to

  • first, using the fact that $A$ has one jordan block, compute the dimension of the space of all matrices of the form $f(A)$;

  • second, compute explicitely the space of matrices which commute with $A$, and determine its dimension

  • observe that the first subspace is contained in the second one, and look at their dimensions.

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