Canonical Forms

Sometimes, a matrix is “ugly”—it is filled with dense numbers that obscure the underlying physics or logic of the system. Canonical forms are the “simplest” possible representations of a linear operator. By changing our basis, we can reveal the true nature of the transformation.

Why Canonical Forms?

If you are studying a physical system, like a vibrating string or a chemical reaction, the equations are often coupled (everything depends on everything else). A canonical form decouples the system.

The most famous canonical form is the Diagonal Form. If a matrix $A$ is diagonalizable, it means there exists a basis where the operator simply scales each axis independently.

The Jordan Normal Form

What happens if a matrix is not diagonalizable? This occurs when there are not enough eigenvectors (the matrix is “defective”).

The Jordan Normal Form (JNF) is the best we can do for any square matrix. It decomposes the operator into Jordan Blocks on the diagonal: $J = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}$

Everything off the block is zero. Inside the block, we have the eigenvalue on the diagonal and 1s just above it. These 1s represent “coupling” that cannot be removed.

Practical Use: Stability Analysis

In control theory, we look at the JNF to determine if a system will explode or settle.

• If the eigenvalues have negative real parts, the system is stable.
• If we have a Jordan block with $\lambda=0$ and a 1 above it, the system might grow linearly over time ($t$), which could be problematic!

import sympy as sp

# Define a 'defective' matrix (not diagonalizable)
A = sp.Matrix([[5, 4, 3], 
             [-1, 0, -3], 
             [1, -2, 1]])

# Compute Jordan Normal Form
P, J = A.jordan_form()

print("Original Matrix A:")
sp.pprint(A)

print("\nJordan Normal Form J:")
sp.pprint(J)

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

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

Rational Canonical Form

The Jordan form requires complex numbers (to find all roots of the characteristic polynomial). If we want to stay within the field of rational numbers $\mathbb{Q}$ or real numbers $\mathbb{R}$, we use the Rational Canonical Form (also known as the Frobenius map).

Instead of eigenvalues, this form uses Invariant Factors—polynomials that divide each other. This is deeply connected to the structure theory of modules over a Principal Ideal Domain (PID).

Exercises