I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze.
Here is the question:
Let $T$ be a a linear operator on an $n$-dimensional space, and suppose that $T$ has $n$ distinct characteristic values. Prove that any linear operator $U$ which commutes with $T$ is a polynomial in $T$.
My work so far: Since $T$ has $n$ distinct characteristic values and since the space it acts on is also $n$ dimensional, $T$ must be diagonalisable. Now using the fact $UT=TU$ I can show that $U$ must also be diagonalizable. So these two operators must be simultaneously diagonalizable. Now I'm stuck with where this is going. I'm thinking of simultaneous diagonalizability because this is section 6.5 of Hoffman and Kunze and it deals with simultaneous diagonalizability. Am I on the right track with this?. Can anybody help?
Thanks so much for your time and your answers.
Best Answer
Note: we will prove that the commutant of $T$ is equal to $K[T]$. This property is actually equivalent to the fact that the characteristic and minimal polynomials are equal. That's also equivalent to similarity to a companion matrix.
Let $\lambda_j$ be the $n$ pairwise distinct eigenvalues of $T$. Such an operator must be diagonalizable, and its eigenspaces are all one-dimensional. Since $U$ commute with $T$, it leaves the eigenspaces of $T$ invariant. Since they are one-dimensional, this implies that every eigenvector of $T$ is an eigenvector for $U$. So $T$ and $U$ are simultaneously diagonalizable.
In a basis of simultaneous diagonalization, $T=\mbox{diag}(\lambda_1,\ldots,\lambda_n)$ and $U=\mbox{diag}(\mu_1,\ldots,\mu_n)$.
Now we will do Lagrange interpolation. Since the $\lambda_j$ are pairwise distinct, we can consider the degree $n$ polynomials $$ L_j(x)=\prod_{i\neq j}\frac{x-\lambda_i}{\lambda_j-\lambda_i}\qquad L_j(\lambda_i)=\delta_{ij}. $$ Now $$p(x)=\sum_{j=1}^n\mu_jL_j(x)\qquad\mbox{satisfies }\quad p(\lambda_j)=\mu_j \;\forall j. $$ Therefore $p(T)=\mbox{diag}(p(\lambda_1),\ldots,p(\lambda_n))=U$ belongs to $K[T]$. The converse is clear, so the commutant of $T$ is $$ \{T\}'=\{U\in L(V)\,;\,UT=TU\}=K[T] $$ the subagebra of all polynomials in $T$.
Note: for a less explicit argument, just compare the dimensions of $K[T]$ and the subspace of all diagonal operators in this basis. The latter is clearly $n$. The former is $n$ by minimal polynomial consideration and Euclidean division. And clearly $K[T]$ is contained in the commutant, so they must be equal. Oh...but that's TTS argument...