Abstract Linear Algebra 38 | Invariant Subspaces

Invariant subspaces are the key structural tool behind Jordan normal form: a subspace U of a vector space V is called invariant under a linear map L when applying L to any vector in U never leaves U. Formally, invariance means L(U) ⊆ U, so the image of U under L is either equal to U or a smaller subspace contained in it. This property matters because it lets mathematicians restrict attention to U itself—turning L into a new linear map that acts only within U—making complicated linear-algebra problems more manageable.

That restriction becomes especially powerful when studying a square matrix A over C. The path toward Jordan normal form starts with eigenvalues: once an eigenvalue Λ is fixed, the generalized eigenspace machinery builds a nested chain of subspaces using powers of (A − ΛI). The generalized eigenspace chain is organized by kernels: for each exponent k, one considers ker((A − ΛI)^k). As k increases, these kernels grow until they stabilize at the fitting index d (also called the size parameter for the chain). The fitting index satisfies 1 ≤ d ≤ n, where n is the matrix size, and different matrices can have different fitting indices.

A parallel chain exists for ranges: the same stabilization phenomenon occurs for the images of (A − ΛI)^k. As the exponent increases, the ranges shrink, again ending at the fitting index d. The crucial payoff is that the final subspaces from both chains—the stabilized kernel and the stabilized range—are invariant under the matrix A. In other words, once the chain has reached its terminal step, those terminal subspaces behave like “closed worlds” under the action of A.

The invariance proof uses the simplification of working with N = A − ΛI instead of A directly. Since A = N + ΛI, applying A to a vector x can be expressed as Ax = Nx + Λx. If both Nx and x lie in a candidate invariant subspace, then Ax automatically lies there too. For the kernel side, take x in ker(N^d). Then N^d x = 0, and applying N gives N^d(Nx) = N(N^d x) = N·0 = 0, so Nx remains in ker(N^d). That shows ker(N^d) is invariant under N, and therefore invariant under A.

For the range side, take y in ran(N^d). By definition of range, y = N^d x for some x. Then Ny = N(N^d x) = N^d(Nx), which is again an element of ran(N^d) because it has the same “N^d times something” form. This establishes invariance of ran(N^d) under N, and hence under A.

These invariant subspaces—built from the stabilized generalized eigenspace kernel and range—set up the decomposition into Jordan blocks. The details of that decomposition are reserved for the next step, but the structural groundwork is now in place: once the fitting index is reached, the terminal kernel and range provide invariant subspaces that can be used to carve the matrix into its Jordan components.