## Annihilating Polynomials

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### Annihilating Polynomials

These are two exercises from Hoffman/Kunze book Linear Algebra, chapter 6, section 3.

First: Let $V$ be an n-dimensional vector space and let $T$ be a linear operator on $V$. Suppose that there exists some positive integer $k$ so that $T^k = 0$. Prove that $T^n = 0$.

Second: Let $A$ be an $n \times n$ matrix with characteristic polynomial $f = (x-c_1)^{d_1} \cdots (x-c_k)^{d_k}$.
Show that $c_1d_1 + \cdots + c_kd_k = \text{trace} (A)$.

These are my solutions, respectively:

If there exists some positive integer such that $T^k = 0$, then the minimal polynomial for $T$ has degree $m$ with $m \leq k$. Since the minimal and characteristic polynomials have the same roots by the Cayley-Hamilton theorem, and that the characteristic polynomial has degree $n$ and also annihilates T, we have $T^n = 0$.

If $A$ has $f$ as characteristic polynomial then it is similar to a triangular matrix of the form $\begin{bmatrix} \begin{bmatrix} c_1 & * & * \\ \vdots & \ddots & * \\ 0 & \cdots & c_1 \end{bmatrix} & \cdots & \cdots & * \\ 0 & \begin{bmatrix} c_2 & * & * \\ \vdots & \ddots & * \\ 0 & \cdots & c_2 \end{bmatrix} & \cdots & * \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \begin{bmatrix} c_k & * & * \\ \vdots & \ddots & * \\ 0 & \cdots & c_k \end{bmatrix} \end{bmatrix}$ where the $c_j$ blocks are $d_j \times d_j$ size.

Thus, the trace of this matrix is $c_1d_1 + \cdots + c_kd_k$ which is the same trace as the original matrix, since if $B = P^{-1} A P$, then $\text{trace}(B) = \text{trace}(P^{-1}AP) = \text{trace}((AP)P^{-1}) = \text{trace}(A)$.

I'm sorry if I can't post two questions in the same thread, in that case please give me a heads up and I shall create a new one, also apologize for the length.

Is my reasoning correct?
Could I have worded it better?
Did I lack information?
In the second problem, is there a way could I do it without using the similarity to the triangular form (given that it's proved in the next chapter that if $F$ is an algebraically closed field then every $n \times n$ matrix is similar to a triangular matrix)?

I'm currently a math bachelor and I had problems with this subject last semester, leading to a flunk (there were professor issues as well) and I know having a great grasp of this is essential for everything that follows, plus I'm really dedicated to do my best and understand this thoroughly.
"There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent."
- Raoul Bott
Fantini
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