$W$ is a $T$-invariant subspace of $V$, prove a Jordan form of $T|_W$ contained the Jordan form of $T$. The meaning of the title is to show that each block of the Jordan form of $T|_W$ corresponds to a block in the Jordan form of $T$ of equal or greater size. I know that each Jordan block in the form of $T|_W$ corresponds to a chain of vectors $\{(T-\lambda I)^k(w),(T-\lambda I)^{k-1}(w),...,w\}$, and that each such chain in $W$ can be lengthened in $V$ since it's possible $w=(T-\lambda I)(v)$ for some $v\notin W$, but I'm not able to show that this chain must correspond to a block in the Jordan form of $T$. This is where I'm stuck.
So, is it true in general that if you find a chain of the above form $$\{(T-\lambda I)^m(v),(T-\lambda I)^{m-1}(v),...,v\}$$ in a vector space, where $(T-\lambda I)^m(v)$ is an eigenvector, then there must be some Jordan block of the eigenvalue of size $m$ in the Jordan form of $T$? And if you find $n$ such chains with independent eigenvectors in each chain, there must be $n$ such blocks?
Thanks.
 A: In general we have $\sigma(T|_W) \subseteq \sigma(T)$ so every eigenvalue $\lambda$ of $T|_W$ is an eigenvalue of $T$. Recall that
$$\dim \ker (T-\lambda I) = \text{number of $\lambda$-blocks}$$
$$\dim \ker (T-\lambda I)^2 - \dim \ker (T-\lambda I) = \text{number of $\lambda$-blocks of size $\ge 2$}$$
$$\dim \ker (T-\lambda I)^3 - \dim \ker (T-\lambda I)^2 = \text{number of $\lambda$-blocks of size $\ge 3$}$$
$$\vdots$$
For any $w \in W$ such that $(T-\lambda I)^{j-1}|_W w \ne 0$ and $(T-\lambda I)^{j}|_W w = 0$ we also have $(T-\lambda I)^{j-1} w \ne 0$ and $(T-\lambda I)^{j} w = 0$ so clearly
\begin{align}
\dim \ker (T-\lambda I)|_W^{j} - \dim \ker (T-\lambda I)|_W^{j-1} &= \dim \left(\ker(T-\lambda I)|_W^{j} \dot{-}\, \ker(T-\lambda I)|_W^{j-1}\right)\\
&\ge \dim \left(\ker(T-\lambda I)^{j} \,\dot{-}\, \ker(T-\lambda I)^{j-1}\right)\\
&= \dim \ker (T-\lambda I)^j - \dim \ker (T-\lambda I)^{j-1} 
\end{align}
Therefore the number of $\lambda$-Jordan blocks of $T|_W$ of size $\ge j$ is less than or equal to the number of $\lambda$-Jordan blocks of $T$ of size $\ge j$.
Hence for every $\lambda$-Jordan block of $T|_W$ there is a $\lambda$-Jordan block of $T$ of greater or equal size.
