Let $V$ be a complex vector space with a linear operator $T : V \to V$ and a $T$-invariant subspace $W \subseteq V.$ Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $T|_W,$ i.e., for each block in the Jordan Canonical Form of $T|_W,$ there fits an equal or larger block in the Jordan Canonical Form of $T.$
This is something we saw in class, and my tutor casually explained it without a formal proof. I used it to solve some other exercises, but I'm interested in seeing a formal proof to convince myself it's true, as I failed to prove it myself.
He mentioned that we can use $\ker(T|_W - \lambda I)^k \subseteq \ker(T-\lambda I)^k,$ and then, using the algorithm for finding a Jordan basis, we can show it's true, but I don't get it.
Crucially, the characteristic polynomial of $T|_W$ divides the characteristic polynomial of $T,$ and the minimal polynomial of $T|_W$ divides the minimal polynomial of $T.$ Consequently, if we obtain the Jordan Canonical Form for $T|_W$ by finding the Smith Normal Form of $xI - T|_W$ and using the elementary divisors, then these elementary divisors are among the elementary divisors obtained from the Smith Normal Form of $xI - T.$ Ultimately, this guarantees that the Jordan Canonical Form for $T|_W$ is a submatrix of the Jordan Canonical Form for $T.$ For an excellent reference on how to obtain the Smith Normal Form $xI - T$ and the elementary divisors of $T,$ see sections 12.2 and 12.3 of the text Abstract Algebra by Dummit and Foote.