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Chapter 44: Weyr Canonical Form

Prerequisites: Jordan Canonical Form (Ch12) · Nilpotent Matrices · Invariant Subspaces (Ch42)

Chapter Outline: Another Choice for Canonical Forms → Definition of Weyr Blocks → Structure of the Weyr Canonical Form (WCF) as a Large Block Diagonal Matrix → Core Concept: The Weyr Characteristic → Duality with Jordan Characteristic (Conjugate Partitions) → Existence and Uniqueness of WCF → Advantages in Commutative Algebra of Operators → Applications: Commutativity Testing, Dimension of Matrix Algebras, and Simultaneous Canonical Forms

Extension: The Weyr Canonical Form is the "dual" of the Jordan Canonical Form; while the Jordan form is more intuitive for differential equations, the Weyr form provides a clearer hierarchical structure for matrix commutativity and advanced problems in algebraic geometry—a powerful tool for investigating the commutants of operators.

In discussions of similarity canonical forms, the Jordan form almost entirely dominates the scene. However, there exists another structure, equally important and superior in certain algebraic aspects: the Weyr Canonical Form (WCF). Unlike the Jordan form, which decomposes space into independent chains, the Weyr form arranges the successive kernels of the generalized eigenspaces "horizontally." This structure makes the relationship between a matrix and its commutant immediately apparent.

44.1 Definition of Weyr Blocks and WCF

Definition 44.1 (Weyr Block)

A Weyr block corresponding to eigenvalue \(\lambda\) is a block upper triangular matrix: $\(W = \begin{pmatrix} \lambda I_{n_1} & E_1 & 0 & \cdots \\ 0 & \lambda I_{n_2} & E_2 & \cdots \\ \vdots & \vdots & \ddots & \ddots \\ 0 & 0 & \cdots & \lambda I_{n_k} \end{pmatrix}\)$ where \(E_i = \begin{pmatrix} I_{n_{i+1}} \\ 0 \end{pmatrix}\) are identity-like blocks of appropriate dimensions. The sequence \((n_1, n_2, \ldots, n_k)\) is called the Weyr Characteristic.

44.2 Weyr vs. Jordan Characteristics

Theorem 44.1 (Duality)

  1. Jordan Characteristic: The partition formed by the sizes of the Jordan blocks.
  2. Weyr Characteristic: The conjugate partition of the Jordan characteristic. Example: If the Jordan blocks have sizes \((2, 1)\), the Weyr characteristic is \((2, 1)\). If the Jordan block is \((3)\), the Weyr characteristic is \((1, 1, 1)\).

44.3 Applications in Commutativity

Commutant Criterion

A matrix \(X\) commutes with a matrix \(A\) in Weyr Canonical Form iff \(X\) is also block upper triangular and satisfies specific compatibility conditions between its blocks. Significance: WCF makes finding common invariant subspaces and computing the dimension of the centralizer algebra extremely straightforward.

Exercises

1. [Basics] Find the Weyr Canonical Form of \(J_3(0)\).

Solution

Steps: 1. \(J_3(0)\) has a single block of size 3. Its partition is \((3)\). 2. Find the conjugate partition: Rotate a row of 3 squares into a column of 3 squares. 3. The result is \((1, 1, 1)\). Construct Weyr Block: \(W = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}\). Conclusion: For a single Jordan block, the Weyr form is identical to the Jordan form.

2. [Calculation] If \(A = \operatorname{diag}(0, 0)\), what is its Weyr characteristic?

Solution

Analysis: 1. The Jordan characteristic is \((1, 1)\). 2. Conjugate partition: Two columns of height 1 combine into one column of height 2. 3. The result is \((2)\). Conclusion: The Weyr characteristic is \((2)\). WCF groups the eigenspace into a single diagonal block.

3. [Structure] Describe the relationship between the dimensions \(n_i\) of the identity blocks \(I_{n_i}\) in a Weyr block.

Solution

Property: Following the properties of Weyr characteristics, \(n_1 \ge n_2 \ge \cdots \ge n_k\). This reflects that the increments in the kernel dimensions \(\ker(A-\lambda I)^i\) are monotonically non-increasing.

4. [Duality] Given Jordan block sizes \((2, 2, 1)\), calculate the Weyr characteristic.

Solution

Calculation: 1. Draw the Young Tableau: Row 1: 2 squares Row 2: 2 squares Row 3: 1 square 2. Count squares in each column: Column 1: 3 squares Column 2: 2 squares Conclusion: The Weyr characteristic is \((3, 2)\).

5. [Uniqueness] Why is the WCF unique?

Solution

Reasoning: 1. The Weyr characteristic is uniquely determined by the sequence of ranks \(\operatorname{rank}(A-\lambda I)^k\). 2. Since rank is an invariant under similarity, the Weyr characteristic is an invariant. 3. Once the order of eigenvalues is fixed, the WCF is perfectly unique.

6. [Commutativity] If \(A\) is in WCF and has distinct eigenvalues, what is the structure of its commutant \(X\)?

Solution

Conclusion: \(X\) must be a diagonal matrix. When eigenvalues are distinct, every Weyr block is \(1 \times 1\), so the whole matrix is diagonal. Any matrix commuting with a diagonal matrix (with distinct entries) is itself diagonal.

7. [Comparison] What is the main morphological difference between Jordan and Weyr forms?

Solution

Core Difference: - Jordan form: Emphasizes "vertical" depth (the length of Jordan chains). - Weyr form: Emphasizes "horizontal" width (the dimension of each generalized eigenspace). WCF groups generalized eigenvectors of the same rank together, making the hierarchical projection structure of the operator clearer.

8. [Nilpotent] Prove: If \(A\) is nilpotent, the diagonal entries of its WCF are all 0.

Solution

Proof: 1. All eigenvalues of a nilpotent matrix are 0. 2. The diagonal blocks of WCF are \(\lambda I_{n_i}\). 3. Substituting \(\lambda = 0\) results in diagonal blocks of zero matrices.

9. [Dimension] If the Weyr characteristic of \(A\) is \((n_1, \ldots, n_k)\), what is the dimension of its commutant algebra?

Solution

Formula: \(\dim \mathcal{C}(A) = \sum_{i=1}^k (2i-1) n_i\). This demonstrates the convenience of the Weyr characteristic for precisely calculating the dimension of an operator's centralizer.

10. [Application] Why is WCF preferred over Jordan form in matrix algebra studies?

Solution

Reasoning: The WCF is block upper triangular with scalar diagonal blocks. This ensures that interactions between blocks follow very simple algebraic rules when dealing with matrix polynomials and equations. It provides a clearer basis for studying the algebraic structures generated by a set of operators.

Chapter Summary

The Weyr Canonical Form is the perfect representation of operator hierarchies:

  1. Conjugate Features: By mapping to the Jordan characteristic via conjugation, WCF proves that matrix similarity classes possess two complementary geometric descriptions, broadening our understanding of operator degeneracy.
  2. Map of Commutativity: The block triangular structure of WCF provides intuitive criteria for matrix commutativity, making it the tool of choice for problems involving families of commuting operators.
  3. Algebraic Clarity: By grouping generalized eigenvectors of the same rank, WCF simplifies the calculation of dimensions for commutants and sub-algebras, occupying an irreplaceable position in advanced representation theory and matrix analysis.