Canonical Forms and the Jordan Normal Form

Diagonalization is the most convenient way to understand a linear operator, as it decomposes the action of a matrix $A$ into independent scaling operations along its eigenvectors. However, many matrices are deficient, meaning they do not possess a complete basis of eigenvectors. The Jordan Canonical Form (JCF) provides the ultimate generalization: it is the “closest” a non-diagonalizable matrix can get to being diagonal.

1. When Diagonalization Fails

A matrix $A\in {\mathbb{C}}^{n\times n}$ is diagonalizable if and only if for every eigenvalue ${\lambda }_i$, the algebraic multiplicity $a_i$ (its multiplicity as a root of the characteristic polynomial $p(x)=\det (xI-A)$) equals its geometric multiplicity $g_i$ (the dimension of the eigenspace $E_{{\lambda }_i}=\ker (A-{\lambda }_{i}I)$).

When $g_i<a_i$, the matrix is deficient. Geometrically, this means the operator “shears” the space in a way that collapses dimensions, preventing the existence of an eigenbasis. To resolve this, we must look beyond the kernels of $(A-\lambda I)$ to the kernels of higher powers $(A-\lambda I{)}^k$.

2. Generalized Eigenvectors and Chains

We define the generalized eigenspace $K_{\lambda }$ associated with $\lambda$ as: $K_{\lambda }={\bigcup }_{k=1}^{\infty }\ker (A-\lambda I{)}^k$ For a matrix of size $n$, this sequence of kernels stabilizes at $k\le a_{\lambda }$. Thus, $K_{\lambda }=\ker (A-\lambda I{)}^{a_{\lambda }}$. The dimension of $K_{\lambda }$ is exactly $a_{\lambda }$.

A Jordan chain of length $k$ is a sequence of vectors $\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_k\}$ such that:

Equivalently, $(A-\lambda I{)}^k{\mathbf{v}}_k=0$ but $(A-\lambda I{)}^{k-1}{\mathbf{v}}_k\ne 0$.

3. The Jordan Block

A Jordan block $J_k(\lambda )$ is a $k\times k$ upper triangular matrix with the eigenvalue $\lambda$ on the diagonal and $1$s on the superdiagonal:

The action of $A$ on the basis formed by a Jordan chain exactly corresponds to the structure of a Jordan block.

4. The Jordan Canonical Form Theorem

Theorem: Let $A$ be an $n\times n$ matrix over an algebraically closed field (like $\mathbb{C}$). Then $A$ is similar to a block diagonal matrix $J$: $A=PJ{P}^{-1},J=\text{diag}(J_{k_1}({\lambda }_1),J_{k_2}({\lambda }_2),\dots ,J_{k_m}({\lambda }_m))$ where $J$ is the Jordan Canonical Form of $A$. This form is unique up to the permutation of the Jordan blocks.

5. Minimal Polynomial and JCF

The minimal polynomial $m_A(x)$ is the unique monic polynomial of least degree such that $m_A(A)=0$. While the characteristic polynomial $p(x)=\prod (x-{\lambda }_i{)}^{a_i}$ tells us the total dimensions of generalized eigenspaces, the minimal polynomial $m_A(x)=\prod (x-{\lambda }_i{)}^{d_i}$ tells us the size of the largest Jordan block for each eigenvalue ($d_i$).

• $A$ is diagonalizable $⟺$ $m_A(x)$ has no repeated roots.
• All Jordan blocks for $\lambda$ are $1\times 1⟺d_{\lambda }=1$.

6. Computing the JCF

Determining the block structure involves calculating the ranks of powers of the shifted matrix $Y=A-\lambda I$. Let $r_k=\text{rank}(Y^k)$. The number of Jordan blocks of size at least $k$ for $\lambda$ is given by: $N_k=r_{k-1}-r_k$ The number of Jordan blocks of exactly size $k$ is: $\text{count}(J_k(\lambda ))=(r_{k-1}-r_k)-(r_k-r_{k+1})=r_{k-1}-2r_k+r_{k+1}$ (setting $r_0=n$).

7. Applications: Differential Equations

For a system $\dot{\mathbf{x}}=A\mathbf{x}$, the solution is $\mathbf{x}(t)=e^{At}\mathbf{x}(0)$. If $A$ is non-diagonalizable, we compute the matrix exponential via the JCF: $e^{At}=P{e}^{Jt}{P}^{-1}$ The exponential of a Jordan block $J_k(\lambda )$ is:

This explains the appearance of terms like $t{e}^{\lambda t}$ in the solutions of ODEs with repeated roots in the characteristic equation.

8. Rational Canonical Form

If the field $\mathbb{F}$ is not algebraically closed (e.g., $\mathbb{Q}$), a matrix might not have JCF over $\mathbb{F}$. The Rational Canonical Form (or Frobenius Normal Form) decomposes the space into cyclic subspaces based on the invariant factors of the matrix, using companion matrices $C(p_i)$ on the blocks. This form works over any field.

Python Implementation: Computing JCF with SymPy

SymPy provides a robust way to compute the Jordan Normal Form and the similarity transform $P$.

import sympy as sp

# Define a non-diagonalizable matrix
A = sp.Matrix([
    [5,  4,  2,  1],
    [0,  1, -1, -1],
    [-1, -1, 3,  0],
    [1,  1, -1,  2]
])

# Compute Jordan Form
# P is the transition matrix, J is the Jordan Form
P, J = A.jordan_form()

print("Jordan Canonical Form (J):")
sp.pprint(J)

print("\nSimilarity Transform Matrix (P):")
sp.pprint(P)

# Verify A = P * J * P^-1
assert A == P * J * P.inv()
print("\nVerification successful: A = PJP^-1")