Jordan canonical form simplifies complex linear transformations into a more manageable structure. It breaks down matrices into blocks, each centered around an eigenvalue, revealing crucial information about the transformation's behavior and properties.

This form is a powerful tool for understanding linear systems, solving differential equations, and analyzing dynamical systems. It provides deep insights into a matrix's structure, making it easier to predict long-term behavior and stability in various applications.

Jordan Canonical Form

Definition and Structure

• Jordan canonical form reveals the structure of a linear transformation with respect to a carefully chosen basis
• Block diagonal matrix composed of Jordan blocks
• Jordan block consists of a square matrix with:
  • Single eigenvalue λ on the main diagonal
  • 1's on the superdiagonal
  • 0's elsewhere
• Size of each Jordan block corresponds to the geometric multiplicity of its associated eigenvalue
• Unique up to the ordering of Jordan blocks
• Every square matrix over an algebraically closed field is similar to a matrix in Jordan canonical form
• Decomposes a linear transformation into its simplest possible form, revealing:
  • Eigenvalues
  • Algebraic multiplicities

Mathematical Properties

• Provides a decomposition of a linear transformation into its simplest possible form
• Reveals both eigenvalues and their algebraic multiplicities
• Algebraic multiplicity equals the sum of sizes of all Jordan blocks for an eigenvalue
• Geometric multiplicity equals the number of Jordan blocks for each eigenvalue
• Jordan blocks of size greater than 1 indicate:
  • Presence of generalized eigenvectors
  • Non-diagonalizable components of the transformation
• 1's on the superdiagonal represent "mixing" of generalized eigenvectors under repeated application of the transformation

Computing Jordan Canonical Form

Eigenvalue Analysis

• Find eigenvalues of the matrix and their algebraic multiplicities
• Determine geometric multiplicity for each eigenvalue by finding the dimension of its eigenspace
• Construct generalized eigenvectors when geometric multiplicity is less than algebraic multiplicity
  • Generalized eigenvector of rank k for eigenvalue λ satisfies: $(\mathbf{A} - \lambda\mathbf{I})^k \mathbf{v} = \mathbf{0}$ but $(\mathbf{A} - \lambda\mathbf{I})^{k-1} \mathbf{v} \neq \mathbf{0}$
• Form Jordan chains by arranging generalized eigenvectors in descending order of rank (longest chain to shortest)

Matrix Transformation

• Construct change of basis matrix P using Jordan chains as columns
• Obtain Jordan canonical form J through similarity transformation: $\mathbf{J} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P}$
• Verify correctness of Jordan form by checking:
  • Block diagonality
  • Jordan block structure
• Example: For a 3x3 matrix with eigenvalue λ = 2 (algebraic multiplicity 3, geometric multiplicity 1): $\mathbf{J} = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}$

Jordan Canonical Form Structure

Interpretation of Components

• Eigenvalues on diagonal represent scaling factors of transformation along principal directions
• Size of each Jordan block corresponds to dimension of largest invariant subspace associated with its eigenvalue
• Number of Jordan blocks for each eigenvalue equals its geometric multiplicity
• Sum of sizes of all Jordan blocks for an eigenvalue equals its algebraic multiplicity
• Example: For a 4x4 matrix with Jordan form: $\mathbf{J} = \begin{bmatrix} 3 & 1 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}$
  • Eigenvalues: 3 (algebraic multiplicity 2, geometric multiplicity 1), 2, and -1
  • Two Jordan blocks: one 2x2 for λ = 3, two 1x1 for λ = 2 and λ = -1

Geometric Interpretation

• Overall structure provides insight into decomposition of vector space into cyclic subspaces invariant under linear transformation
• Jordan blocks of size greater than 1 represent non-diagonalizable components
• Presence of 1's on superdiagonal indicates "mixing" of generalized eigenvectors
• Example: In a 3D space, a Jordan form with a 2x2 block and a 1x1 block represents:
  • A plane where vectors rotate and scale
  • A line where vectors only scale

Jordan Canonical Form for Dynamical Systems

Continuous Systems

• Used to solve systems of linear differential equations $\mathbf{x}' = \mathbf{A}\mathbf{x}$
• General solution expressed as $\mathbf{x}(t) = \mathbf{P}e^{\mathbf{J}t}\mathbf{P}^{-1}\mathbf{x}(0)$
  • J is Jordan form of A
  • P is change of basis matrix
• Exponential of Jordan block computed using truncated Taylor series expansion
• Eigenvalue analysis for solution behavior:
  • Negative real parts lead to decaying solutions
  • Positive real parts lead to growing solutions
• Jordan blocks introduce polynomial factors in solution (terms like $t^k e^{\lambda t}$)
• Example: For a system with Jordan form: $\mathbf{J} = \begin{bmatrix} -2 & 1 \\ 0 & -2 \end{bmatrix}$ Solution will contain terms like $e^{-2t}$ and $te^{-2t}$

Discrete Systems

• Analyze long-term behavior of systems $\mathbf{x}(n+1) = \mathbf{A}\mathbf{x}(n)$
• Behavior determined by $\mathbf{A}^n$, analyzed using Jordan form
• Stability of equilibrium points in nonlinear systems studied by:
  • Linearizing the system
  • Examining Jordan form of Jacobian matrix at equilibrium point
• Example: For a discrete system with Jordan form: $\mathbf{J} = \begin{bmatrix} 0.5 & 0 \\ 0 & -0.8 \end{bmatrix}$ System will converge to equilibrium as n → ∞ (all eigenvalues have magnitude < 1)