Jordan Form

Definition

Jordan Canonical Form is a matrix representation that is “as close to diagonal as possible” for any square matrix. It handles “defective” matrices that do not have $n$ linearly independent eigenvectors.

Why It Matters

Not every matrix is perfectly diagonalizable. Jordan form is the “best possible” simplification for the messy, degenerate systems of the real world, providing the essential mathematical framework for solving complex differential equations in control theory and physics.

Core Concepts

• Jordan Blocks ($J_i$): Sub-matrices along the diagonal with the eigenvalue $\lambda$ on the diagonal and 1s on the super-diagonal. $J_i = \begin{bmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{bmatrix}$

  • How to read: “The Jordan block J i is equal to a three by three matrix with the eigenvalue lambda on the diagonal and ones on the super-diagonal.”
  • Meaning / when to use: When eigenvectors are insufficient, Jordan blocks capture “coupled” generalized eigenvectors that still transform under the same eigenvalue $\lambda$.

• Generalized Eigenvectors: Vectors that satisfy $(A - \lambda I)^k v = 0$. These form the basis for the Jordan Form.

  • How to read: “The quantity A minus lambda times the identity matrix raised to the power k, times the vector v, equals zero.”
  • Meaning: Generalized eigenvectors become ordinary eigenvectors after $k$ applications of $(A - \lambda I)$.

• Existence: Every square matrix $A$ is similar to its Jordan Form $J$: $A = M J M^{-1}$.

  • How to read: “The matrix A is equal to the product of M, the Jordan form J, and the inverse of M.”
  • Meaning: $A$ and $J$ represent the same linear map in different bases; $J$ is the canonical “almost-diagonal” form.

• Uniqueness: The Jordan Form is unique up to the ordering of the blocks.

Connected Concepts

• Eigenvalues and Eigenvectors
• Change Of Basis
• Linear Transformation