95. Jordan Canonical Form
When a matrix is not diagonalizable (geometric multiplicity algebraic multiplicity for some eigenvalue), the Jordan canonical form is the next-best thing — almost diagonal, with ‘s just above the diagonal in certain blocks.
Every square matrix over (or over if all eigenvalues are real) is similar to its Jordan form :
where is block diagonal:
95.1. Jordan blocks
A Jordan block is a upper triangular matrix with on the diagonal and on the superdiagonal:
If has linearly independent eigenvectors for eigenvalue (geometric multiplicity ) and total multiplicity (algebraic multiplicity), the corresponding Jordan structure has blocks for , whose sizes sum to .
95.2. Diagonalizable as a special case
If is diagonalizable, every Jordan block is — i.e., is just diagonal. The Jordan form generalizes diagonalization to handle “missing eigenvectors.”
Example
Eigenvalue with algebraic multiplicity , geometric multiplicity (only eigenvector ). Not diagonalizable.
Jordan form: (already in Jordan form).
Example
Eigenvalues: (algebraic mult , geometric mult ) and (mult ).
Jordan form has one block for and one block for :
95.3. Why it matters
- Matrix exponential , and on Jordan blocks has a known closed form (involving polynomial factors alongside in the time evolution of linear ODEs)
- Solving linear systems of ODEs with repeated eigenvalues produces polynomial-times-exponential terms — directly traceable to non-trivial Jordan blocks
- Generalized eigenvectors — the “missing” basis vectors live in for
95.4. Computation note
Computing the Jordan form numerically is unstable — small perturbations can turn a non-trivial Jordan block into a tight cluster of distinct eigenvalues. For numerical work, prefer the SVD or the Schur form (always available, real or complex, and stable).
95.5. See also
- Diagonalization
- Eigenvectors & Eigenvalues
- Characteristic Polynomial — algebraic multiplicity
- Spectral Theorem — when Jordan blocks are all (symmetric / Hermitian case)