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Jordan Canonical Form Calculator

Transform matrices to Jordan normal form (JCF)

Enter 2x2 Matrix A

Jordan Canonical Form

A = P * J * P-1
J = Jordan form (block diagonal)
J(λ,n) = Jordan block of size n
J(λ,n) = [[λ,1,0,...],[0,λ,1,...],[...]]

Jordan Canonical Form decomposes a matrix into a block diagonal form where each block is a Jordan block with constant diagonal entries (eigenvalues) and 1s on the superdiagonal.

⚠ JCF always exists over complex numbers. Over reals, use real Jordan form with 2x2 blocks for complex eigenvalues.

What is Jordan Canonical Form?

The Jordan Canonical Form (JCF) is a canonical form for matrices that reveals important structural properties. Every square matrix is similar to a unique Jordan matrix, which consists of Jordan blocks on its diagonal.

Jordan Block

J(λ,n) = λ*I + N where N is nilpotent matrix with 1s on superdiagonal.

Generalized Eigenvectors

Vectors satisfying (A-λI)^k v = 0 for some k.

Transition Matrix

P contains generalized eigenvectors as columns.

Uniqueness

JCF is unique up to permutation of Jordan blocks.

Teaching Example: A = [[3,1],[0,3]]
1. Eigenvalues: λ = 3 (algebraic multiplicity 2)
2. Eigenspace dimension = 1 (geometric multiplicity 1)
3. Find generalized eigenvector: (A-3I)v2 = v1
4. J = [[3,1],[0,3]], P = [v1 v2]

Applications

Matrix Powers Differential Equations Control Theory Group Theory Numerical Analysis

Frequently Asked Questions

What is Jordan Canonical Form?▼
Jordan Canonical Form is a block diagonal matrix where each block is a Jordan block J(λ,n) with eigenvalue λ on diagonal and 1s on superdiagonal.
How to find JCF?▼
Find eigenvalues, compute generalized eigenspaces, find Jordan chains, then construct transition matrix P.
Why JCF is useful?▼
JCF simplifies matrix operations: powers, exponentials, and solving linear differential equations.
Jordan block vs diagonal?▼
Diagonal form is special case when all Jordan blocks are 1x1 (matrix is diagonalizable).

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