restriction to an invariant subspace

A subspace W of V is invariant under an operator T if T never carries a vector of W outside W — that is, T(W) is contained in W. When this holds, T can be restricted to W: the rule v -> T(v) makes perfect sense as an operator T|_W : W -> W, because the output is guaranteed to land back in W. Invariance is precisely the condition that lets you study T on a smaller stage.

Invariant subspaces are how big operators get broken into smaller ones. The cleanest source is eigenvectors: any eigenspace is invariant, and so is any span of eigenvectors, any generalized eigenspace, and the kernel and image of any polynomial in T. Finding invariant subspaces is the same problem as understanding the operator's internal structure.

The matrix payoff is block-triangular form. Extend a basis of an invariant subspace W to a basis of all of V. In that basis the matrix of T has a zero block in the lower-left corner: it looks like [A, B; 0, C]. The top-left block A is the matrix of the restriction T|_W. If you are lucky enough to have a complementary invariant subspace too, the off-diagonal B also vanishes and the matrix is block-diagonal — T splits as a direct sum of two smaller operators.

This block structure is the practical engine behind canonical forms. Determinant and characteristic polynomial factor across the diagonal blocks: det T = det A times det C, and char-poly of T is the product of the blocks' char-polys. Peeling off invariant subspaces one at a time, you reduce any operator to manageable pieces.