Jordan Canonical Form — Definition, Formula & Examples

Jordan Canonical Form is a nearly diagonal matrix representation that every square matrix over the complex numbers can be transformed into, where each diagonal block corresponds to an eigenvalue and captures cases where the matrix is not fully diagonalizable.

For any square matrix $A \in \mathbb{C}^{n \times n}$ , there exists an invertible matrix $P$ such that $P^{-1}AP = J$ , where $J$ is a block-diagonal matrix composed of Jordan blocks $J_k(\lambda_i)$ . Each Jordan block is an upper-triangular $k \times k$ matrix with eigenvalue $\lambda_i$ on the diagonal and ones on the superdiagonal. The Jordan form $J$ is unique up to permutation of the blocks.

Key Formula

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}

Where:

$J_k(\lambda)$ = A k×k Jordan block with eigenvalue λ
$\lambda$ = An eigenvalue of the original matrix
$k$ = Size of the Jordan block

How It Works

To find the Jordan Canonical Form of a matrix

A

, you first compute its eigenvalues and their algebraic and geometric multiplicities. If the geometric multiplicity of an eigenvalue equals its algebraic multiplicity, that eigenvalue contributes only

1 \times 1

diagonal blocks. When the geometric multiplicity is strictly less than the algebraic multiplicity, larger Jordan blocks appear, reflecting the existence of generalized eigenvectors. You then find a basis of generalized eigenvectors to construct the invertible matrix

P

that satisfies

P^{-1}AP = J

Worked Example

Problem: Find the Jordan Canonical Form of the matrix

A = \begin{pmatrix} 5 & 1 \\ 0 & 5 \end{pmatrix}

Step 1: Find the eigenvalues. The characteristic polynomial is

(5 - \lambda)^2 = 0

, so

\lambda = 5

with algebraic multiplicity 2.

\det(A - \lambda I) = (5-\lambda)^2 = 0 \implies \lambda = 5

Step 2: Find the geometric multiplicity. Solve

(A - 5I)\mathbf{x} = \mathbf{0}

. Since

A - 5I = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}

, the null space is one-dimensional, so the geometric multiplicity is 1.

\text{geometric multiplicity} = 1 < 2 = \text{algebraic multiplicity}

Step 3: Since the geometric multiplicity (1) is less than the algebraic multiplicity (2), there is a single 2×2 Jordan block.

J = J_2(5) = \begin{pmatrix} 5 & 1 \\ 0 & 5 \end{pmatrix}

Answer: The Jordan Canonical Form is

J = \begin{pmatrix} 5 & 1 \\ 0 & 5 \end{pmatrix}

. In this case,

A

is already in Jordan form.

Why It Matters

Jordan Canonical Form is essential in solving systems of linear differential equations when the coefficient matrix is not diagonalizable. It also appears in control theory and stability analysis, where the structure of Jordan blocks determines the qualitative behavior of dynamical systems.

Common Mistakes

Mistake: Assuming every matrix is diagonalizable and skipping the check for generalized eigenvectors.

Correction: Always compare the geometric multiplicity to the algebraic multiplicity for each eigenvalue. If they differ, the matrix requires Jordan blocks larger than 1×1.