I've been trying to understand the proof that every matrix in a algebraically closed field can be written in JCF with respect to a basis.
It's been hard to me because the reference that I was following was too compact in this proof, and I also don't think it's a trivial proof.
After reading from another references I started to understand better what's going on, but I still have some doubts about some steps in the proof, and I'd appreciate if someone here can answer this doubts.
Let $T:V \rightarrow V$ be a linear transformation where $V$ is a finite vector space with respect to $\mathbb{K}$, a algebraically closed field and $\{\lambda_1, \cdots, \lambda_n\}$ is the set of the distinct eigenvalues of $T$. It follows that:
$$ \ker(T-\lambda_1I) \subset \ker(T-\lambda_1I)^2 \subset \cdots \subset V $$ and since $V$ is finite then there must be a $m$ such that $\ker(T-\lambda_1I)^m = \ker(T-\lambda_1I)^{m+1}$ and so on... I've followed the proof and agree that $$ V = \underbrace{\ker(T-\lambda_1I)^{m_1}}_{W_1} \oplus \underbrace{\text{Im}(T-\lambda_1I)^{m_1}}_{W_2} $$ where $W_1$ and $W_2$ are $T$ invariant. Now the point is:
- If $\overline{T} = T\mid_{W_1} : W_1 \rightarrow W_1$ i.e $T$ restricted to $W_1$ is nilpotent with index $\alpha$, then $\overline{T}^{\alpha} = 0$ and there is a $v \in W_1$ such that $B_i = \{v,\overline{T}v, \cdots, (\overline{T})^{\alpha-1}v\}$ is a basis for $W_1$.
- If $\lambda_1$ is the only eigenvalue of $\overline{T}$ with algebraic multiplicity $m$ then it follows that $\{\lambda_2, \cdots \lambda_n\}$ are eigenvalues for $T\mid_{W_2}$ and we can use the same reasoning for $V = \text{Im}(T-\lambda_2)^{m_2}$ to see that $$ V = \bigoplus_{i=1}^n \underbrace{\ker(T-\lambda_iI)^{m_i}}_{W_i} $$ where each $W_i$ is going to be invariant under the action of $T$, and therefore, the matrix of $T$ with respect to the basis $B=\cup B_i$ of $V$ is going to be in JCF.
That's what I've been able to see and agree with. But the point is that I can't see why the two facts in bold are true... Can someone please help me with that? Because if they are true I can agree with the full proof.
Thank you and any help will be highly appreciated!