Jordan Canonical Form

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 $1$ ′s just above the diagonal in certain blocks.

Every square matrix $𝐴$ over $ℂ$ (or over $ℝ$ if all eigenvalues are real) is similar to its Jordan form $𝐽$ :

𝐴 = 𝑃 𝐽 𝑃^{- 1}

where $𝐽$ is block diagonal:

𝐽 = [\begin{matrix} 𝐽_{𝜆_{1}, 𝑘_{1}} & 0 & \dots & 0 \\ 0 & 𝐽_{𝜆_{2}, 𝑘_{2}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 𝐽_{𝜆_{𝑟}, 𝑘_{𝑟}} \end{matrix}]

95.1. Jordan blocks

A Jordan block $𝐽_{𝜆, 𝑘}$ is a $𝑘 \times 𝑘$ upper triangular matrix with $𝜆$ on the diagonal and $1$ on the superdiagonal:

𝐽_{𝜆, 1} = [\begin{matrix} 𝜆 \end{matrix}]

𝐽_{𝜆, 2} = [\begin{matrix} 𝜆 & 1 \\ 0 & 𝜆 \end{matrix}]

𝐽_{𝜆, 3} = [\begin{matrix} 𝜆 & 1 & 0 \\ 0 & 𝜆 & 1 \\ 0 & 0 & 𝜆 \end{matrix}]

If $𝐴$ has $𝑘$ linearly independent eigenvectors for eigenvalue $𝜆$ (geometric multiplicity $𝑘$ ) and total multiplicity $𝑚$ (algebraic multiplicity), the corresponding Jordan structure has $𝑘$ blocks for $𝜆$ , whose sizes sum to $𝑚$ .

95.2. Diagonalizable as a special case

If $𝐴$ is diagonalizable, every Jordan block is $1 \times 1$ — i.e., $𝐽 = 𝐷$ is just diagonal. The Jordan form generalizes diagonalization to handle “missing eigenvectors.”

Example

𝐴 = [\begin{matrix} 5 & 1 \\ 0 & 5 \end{matrix}]

Eigenvalue $𝜆 = 5$ with algebraic multiplicity $2$ , geometric multiplicity $1$ (only eigenvector $[\begin{matrix} 1 \\ 0 \end{matrix}]$ ). Not diagonalizable.

Jordan form: $𝐽 = [\begin{matrix} 5 & 1 \\ 0 & 5 \end{matrix}] = 𝐴$ (already in Jordan form).

Example

𝐴 = [\begin{matrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]

Eigenvalues: $2$ (algebraic mult $2$ , geometric mult $1$ ) and $3$ (mult $1$ ).

Jordan form has one $2 \times 2$ block for $𝜆 = 2$ and one $1 \times 1$ block for $𝜆 = 3$ :

𝐽 = [\begin{matrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}] = 𝐴

95.3. Why it matters

Matrix exponential $𝑒^{𝐴} = 𝑃 𝑒^{𝐽} 𝑃^{- 1}$ , and $𝑒^{𝐽}$ on Jordan blocks has a known closed form (involving polynomial factors $𝑡^{𝑘}$ alongside $𝑒^{𝜆 𝑡}$ in the time evolution of linear ODEs)
Solving linear systems of ODEs ${\vec{𝑥}}^{'} (𝑡) = 𝐴 \vec{𝑥} (𝑡)$ with repeated eigenvalues produces polynomial-times-exponential terms — directly traceable to non-trivial Jordan blocks
Generalized eigenvectors — the “missing” basis vectors live in $ker ({(𝐴 - 𝜆 𝐼)}^{𝑘})$ for $𝑘 > 1$

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 $1 \times 1$ (symmetric / Hermitian case)