TITLE: What does it mean for a subspace to be stable?
QUESTION [9 upvotes]: I'm looking through a proof for a spectral theorem, but I can't figure out what it means for a subspace to be stable.

$\dots \mathbb{C}v$ is $T$-stable (for some $v$ that is an
  eigenvector of an endomorphism $T: V \to V$, where $V$ is a vector
  space over a field $k$). Then the orthogonal complement $(\mathbb{C}v)^{\perp}$ is also $T$-stable...

REPLY [8 votes]: An alternative equivalent terminology that is possibly more common in the linear algebra literature is that $S$ is $T$-invariant when $T(S) \subseteq S$.