Chapter 13B: λ-matrices and Rational Canonical Form¶

Prerequisites: Polynomial Algebra (Ch00) · Matrix Algebra (Ch02) · Jordan Canonical Form (Ch12) · Quotient Spaces (Ch13A)

Chapter Outline: From Number Fields to Polynomial Rings → Definition of $\lambda$-matrices and Elementary Operations → Smith Normal Form & Uniqueness Theorem → Determinantal Divisors and Invariant Factors → Elementary Divisors → Necessary and Sufficient Conditions for Matrix Similarity (Equivalence of Characteristic Matrices) → Construction of Companion Matrices → The Rational Canonical Form (RCF) Theorem → Relationship with Jordan Canonical Form → Applications: Matrix Analysis over General Fields, Canonical Forms in Control Systems

Extension: The Rational Canonical Form solves the problem of matrix canonical forms over general fields (e.g., $\mathbb{Q}$) without requiring the field to be algebraically closed (e.g., $\mathbb{C}$); it is a concrete application of the structure theorem for modules over a Principal Ideal Domain (PID).

While the Jordan Canonical Form is theoretically elegant, its construction relies on the existence of eigenvalues within the field. When working over the rational field $\mathbb{Q}$, the characteristic polynomial may not be factorable. The Rational Canonical Form (RCF) overcomes this limitation by providing a universal matrix standard form applicable to any field through the decomposition of polynomial rings. This chapter uses the elementary transformations of $\lambda$-matrices to reveal this profound algebraic construction.

13B.1 λ-matrices and Smith Normal Form¶

Definition 13B.1 (λ-matrix)

A matrix whose entries are polynomials in $\lambda$ is called a $\lambda$-matrix. The characteristic matrix $\lambda I - A$ is the most quintessential example.

Theorem 13B.1 (Smith Normal Form)

Every $n \times n$ $\lambda$-matrix $A(\lambda)$ can be transformed via elementary operations into a unique diagonal form: $$S(\lambda) = \operatorname{diag}(d_1(\lambda), d_2(\lambda), \ldots, d_r(\lambda), 0, \ldots, 0)$$ where each $d_i(\lambda)$ is a monic polynomial such that $d_i(\lambda) \mid d_{i+1}(\lambda)$. These polynomials are called the Invariant Factors of $A(\lambda)$.

13B.2 Similarity and Elementary Divisors¶

Theorem 13B.2 (Similarity Criterion)

Two $n \times n$ matrices $A$ and $B$ are similar iff their characteristic matrices $\lambda I - A$ and $\lambda I - B$ have the same Smith Normal Form (i.e., identical invariant factors).

13B.3 Companion Matrices and Rational Canonical Form¶

Definition 13B.2 (Companion Matrix $C(p)$)

For a monic polynomial $p(\lambda) = \lambda^k + a_{k-1}\lambda^{k-1} + \cdots + a_0$, the companion matrix is defined as: $$C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{k-1} \end{pmatrix}$$ Property: The characteristic and minimal polynomials of $C(p)$ are both $p(\lambda)$.

Theorem 13B.3 (Rational Canonical Form)

Every square matrix $A$ is similar to a block diagonal matrix where each block is the companion matrix of an invariant factor: $$R = \operatorname{diag}(C(d_1), C(d_2), \ldots, C(d_k))$$ This is known as the Rational Canonical Form of $A$. Note that the degree of $d_i$ increases with $i$.

Exercises¶

1. [Smith Form] Calculate the invariant factors of $\begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}$.

Solution

Steps: 1. Determinantal Divisors: - $D_1(\lambda) = \gcd(\lambda, 1, 0, \lambda) = 1$. - $D_2(\lambda) = \det = \lambda^2$. 2. Invariant Factors: - $d_1(\lambda) = D_1 = 1$. - $d_2(\lambda) = D_2 / D_1 = \lambda^2$. Conclusion: The invariant factors are $1, \lambda^2$.

2. [Similarity] If $A$ and $B$ have the same characteristic polynomial $\lambda^2$, are they necessarily similar?

Solution

Conclusion: Not necessarily. Analysis: - Identical characteristic polynomials only mean the product of the invariant factors is the same. - Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$; its invariant factors are $1, \lambda^2$. - Let $B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$; its invariant factors are $\lambda, \lambda$. - Since the sequences of invariant factors differ, they are not similar. This reflects the difference in Jordan block structures.

3. [Companion] Write the companion matrix for $p(\lambda) = \lambda^2 - 3\lambda + 2$.

Solution

Construction: By definition, $a_1 = -3, a_0 = 2$. The companion matrix is $\begin{pmatrix} 0 & -a_0 \\ 1 & -a_1 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 1 & 3 \end{pmatrix}$. Verification: The eigenvalues are 1 and 2, which are the roots of the polynomial.

4. [Minimal Polynomial] In the Rational Canonical Form, which block corresponds to the minimal polynomial?

Solution

Conclusion: The companion matrix block corresponding to the last non-trivial invariant factor $d_k(\lambda)$. Reasoning: In the Smith form, $d_i \mid d_{i+1}$, so $d_k$ contains the highest powers of all elementary divisors, which is the definition of the minimal polynomial.

5. [Divisibility] Prove $d_i(\lambda) \mid d_{i+1}(\lambda)$.

Solution

Logic: 1. Invariant factors are defined as $d_i = D_i / D_{i-1}$, where $D_i$ is the GCD of all $i \times i$ minors. 2. Any $(i+1) \times (i+1)$ minor is a linear combination of $i \times i$ minors. 3. Thus, $D_i$ must divide $D_{i+1}$. 4. Advanced algebraic lemmas prove the quotient sequence also satisfies this divisibility chain.

6. [Contrast] What is the primary difference between Jordan and Rational Canonical Forms?

Solution

Key Differences: 1. Field Dependency: Jordan form requires the characteristic polynomial to factor completely within the field (usually $\mathbb{C}$); RCF exists and is unique over any field (e.g., $\mathbb{Q}, \mathbb{R}$). 2. Decomposition Depth: Jordan form decomposes the space into irreducible linear factor powers (eigenspaces); RCF decomposes into companion blocks of irreducible polynomials (cyclic spaces).

7. [Elementary Divisors] If the invariant factors are $1, (\lambda-1)(\lambda-2)$, what are the elementary divisors?

Solution

Calculation: Elementary divisors are the powers of irreducible factors obtained by factoring the invariant factors over an algebraically closed field. Factoring $(\lambda-1)(\lambda-2)$ yields $(\lambda-1)$ and $(\lambda-2)$. Conclusion: The elementary divisors are $(\lambda-1)$ and $(\lambda-2)$.

8. [Rank] In the Smith form of $\lambda I - A$, why is the number of non-zero diagonal entries $r$ always $n$?

Solution

Conclusion: $r = n$. Reasoning: The determinant of the characteristic matrix $\lambda I - A$ is a polynomial of degree $n$, which is not identically zero. Therefore, the $\lambda$-matrix is of full rank, and must have $n$ non-zero diagonal entries in its Smith form.

9. [Calculation] Find the invariant factors of $J_2(\lambda_0)$.

Solution

Analysis: $A = \begin{pmatrix} \lambda_0 & 1 \\ 0 & \lambda_0 \end{pmatrix} \implies \lambda I - A = \begin{pmatrix} \lambda-\lambda_0 & -1 \\ 0 & \lambda-\lambda_0 \end{pmatrix}$. 1. $D_1 = \gcd(\lambda-\lambda_0, -1, 0, \lambda-\lambda_0) = 1$. 2. $D_2 = (\lambda-\lambda_0)^2$. Conclusion: The invariant factors are $1, (\lambda-\lambda_0)^2$.

10. [Application] Why is RCF important in computational algebra?

Solution

Significance: 1. Avoiding Approximations: Finding eigenvalues involves root-finding, which introduces precision loss. Computing the RCF only requires exact polynomial arithmetic. 2. Field Universality: For matrices where exact eigenvalues cannot be found (e.g., over $\mathbb{Q}$), the RCF provides the only unique, exact way to describe the matrix structure.

Chapter Summary¶

The Rational Canonical Form provides a universal characterization of matrix similarity classes over any field:

Field Independence: By utilizing companion blocks of irreducible polynomials, RCF eliminates the dependency on the complex field, becoming the core of operator theory in abstract algebra.
Polynomial Logic: The theory of invariant factors reveals the deep module structure behind the characteristic matrix $\lambda I - A$, establishing the ultimate algebraic criterion for similarity.
Computational Precision: Compared to the instability of eigenvalue calculation, the construction of Smith forms based on elementary operations provides a robust algorithmic foundation for exact algebra software.