Jordan Canonical Form of a Matrix: Explained with Examples

The Jordan canonical form is the closest a square matrix can get to being diagonal: every square matrix over the complex numbers is similar to a block-diagonal matrix whose blocks have a single eigenvalue on the diagonal and ones just above it. It is the ultimate classification of matrices up to similarity.

It tells you exactly when diagonalization fails and what replaces it, controlling the long-term behavior of linear differential systems and the matrix exponential.

Key Concepts

• Jordan block. J_k(λ) has λ on the diagonal, ones on the superdiagonal, zeros elsewhere; e.g. J_2(3) = [[3, 1], [0, 3]].
• Algebraic multiplicity. How many times λ roots the characteristic polynomial.
• Geometric multiplicity. n − rank(A − λI); equals the number of Jordan blocks for λ.
• Generalized eigenvector. v with (A − λI)^m v = 0, forming chains that fill each block.
• Diagonalizable case. Geometric equals algebraic multiplicity for every eigenvalue, i.e. all blocks are 1 × 1.

Formulas

• Existence: over an algebraically closed field, A = PJP^−1 with J block-diagonal.
• Block sizes: determined by dim ker(A − λI)^m for m = 1, 2, ….
• Matrix exponential: e^{J_k(λ)t} = e^{λt} times a polynomial matrix, which is why defective matrices produce t·e^{λt} terms.
• Assumptions and limits: guaranteed over C; over R complex pairs need real 2 × 2 blocks. Unique up to block ordering.

Worked Example

Find the Jordan form of A = [[2, 1, 0], [0, 2, 0], [0, 0, 3]].

1. Characteristic polynomial: det(A − λI) = (2 − λ)²(3 − λ). Eigenvalues: λ = 2 with algebraic multiplicity 2, λ = 3 with algebraic multiplicity 1.
2. For λ = 2: A − 2I = [[0, 1, 0], [0, 0, 0], [0, 0, 1]]. This matrix has rank 2 (two independent nonzero rows), so nullity = 3 − 2 = 1. Geometric multiplicity is 1, smaller than the algebraic multiplicity 2.
3. Because there is only one eigenvector direction for λ = 2, both copies of the eigenvalue belong to a single Jordan block of size 2: J_2(2).
4. For λ = 3 the multiplicity is 1, so it contributes a 1 × 1 block J_1(3) = [3].
5. Jordan form: J = [[2, 1, 0], [0, 2, 0], [0, 0, 3]]. Here A was already in Jordan form; in general A = PJP^−1 for a suitable P built from (generalized) eigenvectors.
6. Conclusion: A is not diagonalizable, since for λ = 2 the geometric multiplicity 1 differs from the algebraic multiplicity 2.

Common Misconceptions

• Thinking every matrix diagonalizes. Repeated eigenvalues can be defective; the Jordan form is the replacement.
• Expecting a real Jordan form always. The clean triangular form is guaranteed over C; over R complex pairs need real blocks.
• Mixing up the multiplicities. Only the gap between algebraic and geometric multiplicity reveals blocks larger than 1 × 1.

Real-World Applications

• Linear ODE systems: solving x′ = Ax uses the Jordan form; a size-2 block for λ produces solution terms like t·e^{λt}.
• Control theory: the Jordan structure at unstable eigenvalues determines how hard a system is to stabilize.
• Markov chains: the Jordan form of the transition matrix governs convergence rates to the stationary distribution.

The Jordan canonical form writes every complex square matrix as block-diagonal with eigenvalue blocks carrying ones on the superdiagonal. The gap between algebraic and geometric multiplicity is exactly what forces blocks larger than 1 × 1, and those blocks explain the t·e^{λt} terms in differential equations.