invariant subspace

Picture an operator T moving every vector around the space. A subspace W is invariant if T never throws any vector of W outside W: feed in something from W and you always get back something in W. The subspace is sealed against T.

Precisely, W is a subspace of V and T(W) is contained in W, meaning T(w) lies in W for every w in W. Then the restriction T|W is itself an operator on W, so you can study T on the smaller stage W in isolation. Eigenspaces, the kernel, and the image are all invariant; the whole space V and the zero subspace are the trivial ones.

This is the master idea behind every canonical form. Splitting V into invariant pieces is exactly what lets a matrix become block-diagonal or triangular: each invariant subspace becomes a block. A caveat — being invariant says only that T maps W into W, not onto W, and it says nothing about whether a complementary invariant piece exists.