What does it mean to act diagonalisably on a subspace?

Question

Basic question but I'm confused by some text and I can't find a precise definition;

If $f$ is an operator on a vector space $V$, and if $H$ is a subspace of $V$. What does is it mean in precise terms that $f$ acts diagonalisably on $H$?

Thanks!

Answer 1

In general if $f$ is a linear operator on a vector space $W$ then $f$ acts diagonalisably on $W$ if there exists a basis $\mathcal B$ for $W$ such that the matrix of $f$ with respect to the basis $\mathcal B$ is diagonal - that is, there exist scalars $\lambda_b$ such that $f(b) = \lambda_b b$ for each $b \in \mathcal B$.

If $H$ is a subspace of a vector space $V$ then for $f$ to act diagonalisably on $H$ then we need two things: one is that $f(H) \subset H$, so $f|_H$ is a linear operator on the vector space $H$; the second is that $f|_H$ acts diagonalisably on $H$ as defined above.

Answer 2

It means that $f$ restricted to $H$ is a diagonalizable linear transformation (i.e., $f|_H$ has a basis of eigenvectors). This really only makes sense if $f$ maps $H$ to $H$.