Diagonalizabilty of a restriction implies diagonalizability of original operator

Question

I'm stuck trying to solve this problem.

V is finite dimensional vector space over the complex numbers. Suppose $L: V \longrightarrow V$ is linear, and furthermore suppose$U \subset V$ subspace such that $U \subset \ker(L)$ and $im(L)\cap U=\{0\}$. Show that if the induced linear map $L': V/U \longrightarrow V/U $ is diagonalizable, then L is diagonalizable.

First I have tried to consider $L'$ and its basis consisting of eigenvectors such that it is diagonal, also it's minimal polynomial $f(t)=(t-\lambda_1)\cdots(t-\lambda_k)$. Then I can write $L$ in this basis as a block matrix where
L in the given basis is

\begin{bmatrix} L' & B & \\ 0 & C & \\ \end{bmatrix} Still, I am not sure how to proceed. Do I need to show that $U$ is invariant under $L$? Thank you

This is not the same as the problem where you are given a diagonal operator and trying to show that its restriction is diagonal.

Answer 1

If $L'$ is diagonalizable then its minimal polynomial $P(t)$ is a product of distinct linear factors, and for all $v\in V$ we have $(P(L))(v)\in U$. Distinguish two cases:

1. If $P(0)\neq0$ then $tP(t)$ is a product of distinct linear factors, and $L(P(L))(v)=0$ for all $v\in V$ because $(P(L))(v)\in U\subset\ker L$, so the minimal polynomial of $L$ divides $tP(t)$.

2. If $P(0)=0$ then $(P(L))(v)\in\operatorname{im}(L)$ for all $v\in V$. From $\operatorname{im}(L)\cap U=0$ it follows that $P(L)=0$, so the minimal polynomial of $L$ divides $P(t)$.

Answer 2

This is immediate once you reformulate the question a bit. Since $U$ intersects the image of $L$ trivially, one can extend that image to a subspace$~W$ complementary to$~U$. Then choosing some basis according to the decomposition $V=U\oplus W$, the matrix of $L$ with respect to this basis has the block form $$ M=\pmatrix{0&0\\0&D}, $$ since $L(U)=0$ and $L(W)\subseteq W$. The quotient map $V\to V/U$ restricted to$~W$ gives an isomorphism $W\to V/U$, and taking the image of the basis of $W$ as basis of $V/U$, the matrix of $L'$ on this basis is$~D$.

So we are given that $D$ is diagonalisable and asked to prove that $M$ is diagonalisable, which is pretty obvious (a block diagonal matrix built from diagonalisable blocks, here $0$ and $D$, is diagonalisable).