Abstract Linear Algebra 40 | Block Diagonalization

TL;DR

If V = U1 ⊕ U2 and both U1 and U2 are invariant under L, then L is representable by a block diagonal matrix in a suitable basis.

Briefing Cornell Notes

Briefing

A linear map that respects a direct-sum decomposition of the space can always be represented by a block diagonal matrix—no mixing between the two parts. Concretely, if a finite-dimensional vector space V splits as V = U1 ⊕ U2 and both subspaces are invariant under a linear map L (meaning L(U1) ⊆ U1 and L(U2) ⊆ U2), then there exists a basis of V in which the matrix of L has the form

[ A 0 ] [ 0 B ]

with a block size determined by dim(U1) and dim(U2). The off-diagonal blocks are forced to be zero precisely because vectors starting in U1 never leave U1 under L, and similarly for U2.

The proof strategy is constructive: choose a basis for U1 and a basis for U2, then combine them into a basis B for V using the direct-sum property. With respect to this basis, the invariance conditions translate into strong restrictions on how L acts on basis vectors. When the basis isomorphism identifies the chosen basis with standard coordinate vectors in F^n, applying L to vectors corresponding to U1 lands entirely within the coordinates associated to U1, producing zeros in the coordinates tied to U2; the same reasoning applies in the other direction. That coordinate separation is exactly what yields the block diagonal structure, and the block dimensions follow automatically: the first square block is dim(U1) × dim(U1), and the second is dim(U2) × dim(U2).

This block diagonalization is then positioned as a key stepping stone toward Jordan normal form. For a complex square matrix A and an eigenvalue λ, define N = A − λI. The generalized eigenspace for λ is given by ker(N^D), where D is the fitting index (the smallest power for which the kernels stabilize). Prior results establish two crucial facts: ker(N^D) is invariant under A, and the corresponding range space (or complementary invariant subspace built from the image of N^D) is also invariant. Moreover, these two invariant pieces form a direct sum that equals the whole space under consideration.

Once those hypotheses match the earlier block diagonalization proposition, A becomes similar to a 2×2 block diagonal matrix (with respect to a suitable basis). While A itself may not be diagonal, it can be transformed into a matrix whose only nonzero entries lie within two invariant blocks. The first block’s size is tied to dim(ker(N^D)), which is the dimension of the generalized eigenspace. Finally, the characteristic polynomials of A and of the resulting block diagonal matrix coincide—an equality that is explicitly flagged as the fact needed for the next stage of the Jordan normal form argument.

Cornell Notes

If V decomposes as a direct sum V = U1 ⊕ U2 and a linear map L leaves each subspace invariant (L(U1) ⊆ U1 and L(U2) ⊆ U2), then L has a block diagonal matrix representation in some basis. The blocks have sizes dim(U1)×dim(U1) and dim(U2)×dim(U2), and the off-diagonal blocks are zero because vectors cannot “move” between invariant subspaces under L. This framework is then applied to complex matrices: for an eigenvalue λ of A, set N = A − λI and consider the generalized eigenspace ker(N^D), where D is the fitting index. Using invariance and direct-sum properties of ker(N^D) and its complementary invariant subspace, A is shown to be similar to a 2×2 block diagonal matrix. The characteristic polynomial of A matches that of the block diagonal form, setting up the Jordan normal form proof.

Why does invariance of U1 and U2 force the off-diagonal blocks to be zero in the matrix of L?

If L(U1) ⊆ U1, then any vector written using only the basis vectors for U1 stays within the U1 coordinates after applying L. In the matrix representation relative to a basis formed by concatenating bases of U1 and U2, that means columns corresponding to U1 basis vectors have no components in the U2 coordinate directions—so the block in the U1→U2 position is zero. The same argument for L(U2) ⊆ U2 kills the other off-diagonal block.

How is the basis chosen to obtain the block diagonal form?

Pick a basis for U1 of size α = dim(U1), then pick a basis for U2 of size n − α = dim(U2). Because V = U1 ⊕ U2, combining these bases yields a basis B for V. Under the associated basis isomorphism, basis vectors from U1 correspond to standard coordinate vectors with zeros in the “U2 part,” and invariance ensures L maps them back into the same coordinate region.

What are the block sizes in the block diagonal matrix representation?

Let α = dim(U1). The first block corresponds to the action of L restricted to U1, so it is an α×α square block. The second block corresponds to the action on U2, so it is (n−α)×(n−α). The dimensions add because dim(V) = dim(U1) + dim(U2).

How do generalized eigenspaces fit the earlier block diagonalization setup?

For a complex square matrix A and eigenvalue λ, define N = A − λI. The generalized eigenspace is ker(N^D), where D is the fitting index. Known results give that ker(N^D) is invariant under A, and the complementary invariant subspace built from the range of N^D is also invariant. Those two invariant subspaces form a direct sum equal to the whole space, matching the proposition’s assumptions.

What does similarity to a 2×2 block diagonal matrix buy in the Jordan normal form path?

It shows that A can be transformed into a matrix with two invariant blocks rather than a fully diagonal one. The first block’s size is dim(ker(N^D)), linking block structure to generalized eigenspace dimension. Also, the characteristic polynomial of A equals the characteristic polynomial of the block diagonal matrix, which is the ingredient highlighted for the next step toward Jordan form.

Review Questions

Given V = U1 ⊕ U2 and L(U1) ⊆ U1, L(U2) ⊆ U2, what must happen to the coordinates of L(v) when v lies entirely in U1?
In the Jordan normal form context, why is ker((A−λI)^D) invariant under A, and how does that invariance relate to block diagonalization?
How does dim(ker((A−λI)^D)) determine the size of the corresponding block in the similar block diagonal matrix?

Key Points

1
If V = U1 ⊕ U2 and both U1 and U2 are invariant under L, then L is representable by a block diagonal matrix in a suitable basis.
2
The off-diagonal blocks are zero because invariance prevents vectors from moving between U1 and U2 under L.
3
Choosing a basis for U1 and a basis for U2 and concatenating them produces the basis that reveals the block structure.
4
For A over C and eigenvalue λ, setting N = A − λI and using ker(N^D) (with fitting index D) provides an invariant subspace.
5
Generalized eigenspace and its complementary invariant subspace form a direct sum, enabling block diagonalization of A up to similarity.
6
The characteristic polynomial of A matches the characteristic polynomial of the resulting 2×2 block diagonal matrix, supporting the Jordan normal form argument.

Highlights

Invariant subspaces turn abstract linear maps into concrete block diagonal matrices with forced zeros off the diagonal.

The block sizes come directly from dim(U1) and dim(U2), not from any additional algebraic computation.

For A and eigenvalue λ, the generalized eigenspace ker((A−λI)^D) is the key invariant piece whose dimension controls the first block size.

Even when A cannot be diagonalized, it can be made block diagonal up to similarity, and characteristic polynomials stay unchanged.

Topics

Block Diagonalization
Invariant Subspaces
Direct Sum Decomposition
Generalized Eigenspaces
Jordan Normal Form