Computing Jordan Canonical Form

Computing the Jordan form requires finding generalized eigenvectors systematically. The process involves analyzing the kernel structure of powers of $(A - \lambda I)$ .

DefinitionJordan Chain

A Jordan chain for eigenvalue $\lambda$ is a sequence of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}$ satisfying: $(A - \lambda I)\mathbf{v}_1 = \mathbf{0}$ $(A - \lambda I)\mathbf{v}_i = \mathbf{v}_{i-1} \quad \text{for } i = 2, \ldots, k$

The vector $\mathbf{v}_k$ is a generalized eigenvector of rank $k$ , and the chain corresponds to a Jordan block $J_k(\lambda)$ .

ExampleFinding Jordan Form

Let $A = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ .

Step 1: Find eigenvalues. Since $A$ is triangular, eigenvalues are diagonal entries: $\lambda_1 = 2$ (mult. 2), $\lambda_2 = 3$ (mult. 1).

Step 2: For $\lambda = 2$ , compute $\ker(A - 2I)$ : $A - 2I = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ Only one eigenvector, so geometric multiplicity is 1 < 2 = algebraic multiplicity.

Step 3: Find generalized eigenvector in $\ker((A-2I)^2)$ .

Conclusion: Jordan form is $J = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ (which equals $A$ itself—already in Jordan form).

DefinitionGeometric vs Algebraic Multiplicity

For eigenvalue $\lambda$ :

Geometric multiplicity $g_\lambda = \dim(\ker(A - \lambda I))$ counts independent eigenvectors
Algebraic multiplicity $a_\lambda$ is the multiplicity of $\lambda$ as a root of the characteristic polynomial

Always: $1 \leq g_\lambda \leq a_\lambda$

The number of Jordan blocks for $\lambda$ equals $g_\lambda$ . The sum of sizes of these blocks equals $a_\lambda$ .

TheoremStructure of Generalized Eigenspaces

For eigenvalue $\lambda$ with algebraic multiplicity $a_\lambda$ : $\dim(G_\lambda) = \dim(\ker((A - \lambda I)^{a_\lambda})) = a_\lambda$

The dimensions form an increasing chain: $\dim(\ker(A - \lambda I)) \leq \dim(\ker((A - \lambda I)^2)) \leq \cdots \leq \dim(\ker((A - \lambda I)^{a_\lambda})) = a_\lambda$

Remark

Computing Jordan form is more involved than diagonalization and numerically unstable (small perturbations can change the structure). In practice, Jordan form is primarily a theoretical tool for understanding matrix structure. For computation, Schur decomposition is preferred as it's numerically stable.