Restriction of an endomorphism to an invariant subspace

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism. Let $W$ be a $T$-invariant subspace of $V$. The restriction of $T$ to $W$ is the endomorphism $$T|_W \colon W \to W, \qquad T|_W(w) = T(w).$$

Remarks

The map is well defined because $T(W) \subseteq W$, and it is linear. If $V$ is finite-dimensional, then the characteristic polynomial of $T|_W$ divides that of $T$. Take a basis of $V$ that extends a basis of $W$. In this basis, the matrix of $T$ is block upper triangular, and its first diagonal block is the matrix of $T|_W$.

Explanations

L1 human

If a map keeps a subspace inside itself, then it is also a map of that subspace. This smaller map is the restriction. Its eigenvalues are some of the eigenvalues of the original map.