Polynomials and Jordan form

Question

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 $f$.

Answer 1

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.