Jordan Block Construction

• Jordan Block Construction is a method that defines canonical matrix segments using nilpotent matrices to represent eigenvalues in block-diagonal forms.
• It employs an efficient algorithm based on Krylov subspace methods, invariant factors, and minimal annihilator computations to determine block sizes and multiplicities.
• Recent advances show significant performance improvements by reducing reliance on algebraic field extensions and leveraging early termination techniques.

A Jordan block is a canonical matrix segment associated to a single eigenvalue, central to the decomposition of finite-dimensional linear operators. Jordan block construction refers to both the mathematical description and algorithmic computation of the sizes and multiplicities of these blocks in block-diagonal matrix forms, particularly important for explicit representation and analysis of linear maps over rational numbers or integers. Recent advances have improved the efficiency and exactness of calculating Jordan structures for matrices, particularly by leveraging rational canonical form, invariant factors, and annihilator polynomials (Tajima et al., 3 Oct 2025).

1. Mathematical Structure and Foundations

A Jordan block of size $\ell$ for eigenvalue $\alpha \in K$ (where $K = \mathbb{Q}$ or $\mathbb{Z}$) takes the form $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$, with $N_{\ell}$ the nilpotent matrix given by:

$$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$

A Jordan chain of length $\ell$ for a matrix $A\in K^{n \times n}$ and eigenvalue $\alpha$ is a sequence $\alpha \in K$0 in $\alpha \in K$1 such that $\alpha \in K$2, $\alpha \in K$3. The minimal polynomial $\alpha \in K$4 is the monic generator of the ideal $\alpha \in K$5 in $\alpha \in K$6.

For $\alpha \in K$7 with characteristic polynomial $\alpha \in K$8, its invariant factors $\alpha \in K$9 yield the rational canonical form: a block-diagonal matrix with companion blocks $K = \mathbb{Q}$0. Over a splitting field, each $K = \mathbb{Q}$1 further decomposes into Jordan blocks. The exponent of $K = \mathbb{Q}$2 in each $K = \mathbb{Q}$3 dictates the possible block sizes for $K = \mathbb{Q}$4.

The multiplicity $K = \mathbb{Q}$5 of Jordan blocks of size $K = \mathbb{Q}$6 for $K = \mathbb{Q}$7 arises as $K = \mathbb{Q}$8, with $K = \mathbb{Q}$9 the exponent of $\mathbb{Z}$0 in factor $\mathbb{Z}$1. The multiplicity of size exactly $\mathbb{Z}$2 is $\mathbb{Z}$3. Alternatively, computing $\mathbb{Z}$4 gives the count as $\mathbb{Z}$5 for blocks of size at least $\mathbb{Z}$6.

2. Algorithmic Phases for Exact Jordan Block Construction

An efficient algorithm for exact Jordan block structure determination involves three main computational phases, emphasizing avoidance of algebraic field extensions when possible (Tajima et al., 3 Oct 2025):

1. Phase I: Krylov Generating Set via Minimal Annihilators For each standard basis vector $\mathbb{Z}$7, compute $\mathbb{Z}$8, then obtain the minimal annihilator $\mathbb{Z}$9 with $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$0. Vectors $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$1 satisfying $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$2 are used to form a $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$3-basis $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$4 for the relevant generalized eigenspace.
2. Phase II: Extended Krylov Generating Set For $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$5, determine maximal $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$6 such that $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$7, assign $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$8, and store $J_{\ell}(\alpha) = \alpha I_{\ell} + N_{\ell}$9 with $N_{\ell}$0. The set $N_{\ell}$1 comprises these "extended Krylov generators," partitioned by rank.
3. Phase III: Rank-by-Rank Jordan-Krylov Elimination For $N_{\ell}$2 down to $N_{\ell}$3, process each $N_{\ell}$4 in $N_{\ell}$5, performing independence checks and reductions to determine the count $N_{\ell}$6 of size-$N_{\ell}$7 blocks. The process is adaptive: if $N_{\ell}$8 (the undetermined multiplicity) reaches $N_{\ell}$9 or $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$0, the algorithm terminates early.

The condensed pseudocode is:

$\alpha \in K$06

This approach avoids solving linear systems over high-degree algebraic extensions and gains significant efficiency benefits from early termination and rational arithmetic.

3. Role of Invariant Factors and Dimension Formulas

Block size distribution for each eigenvalue $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$1 can be derived from the factorization of invariant factors $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$2, specifically the exponents of $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$3, and is formalized as:

• $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$4
• Blocks of size exactly $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$5 at $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$6 are $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$7

Alternatively, dimension counts $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$8 yield:

• Number of blocks of size $$N_\ell = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & \ddots & \ddots & \ 0 & \cdots & 0 & 0 \end{bmatrix}$$9 is $\ell$0
• Number of blocks of size exactly $\ell$1 is $\ell$2

Computations involving minimal annihilators use GCD algorithms over $\ell$3 (often Euclidean).

4. Computational Complexity and Efficiency

The total computational effort is largely determined by three parameters: $\ell$4, $\ell$5, and $\ell$6 irreducible factors of $\ell$7. Complexity analysis yields:

• Phase I (minimal annihilators): $\ell$8 deterministic, or $\ell$9 randomized.
• Phases II and III: $A\in K^{n \times n}$0 via matrix multiplications and reductions.

Overall, $A\in K^{n \times n}$1 bit-operations randomized or $A\in K^{n \times n}$2 deterministic. By comparison, full Jordan chain methods (solving over degree-$A\in K^{n \times n}$3 extensions) require $A\in K^{n \times n}$4 bit-operations. Reduction to $A\in K^{n \times n}$5-arithmetic and rapid detection of block structure yield substantial practical savings, especially when $A\in K^{n \times n}$6 is large or $A\in K^{n \times n}$7 (Tajima et al., 3 Oct 2025).

5. Observed Performance in Numerical Experiments

Empirical benchmarks on matrices up to $A\in K^{n \times n}$8 over $A\in K^{n \times n}$9, with varying Jordan structure and irreducible factors, indicate:

• Phase I dominates for matrices with many irreducible factors, with reported $\alpha$0–$\alpha$1 reduction in minimal-annihilator cost.
• Phase III, when implemented with "matrix-form" (batch) elimination, is $\alpha$2–$\alpha$3 faster than single-vector methods.
• Overall, speedups over naive Jordan-Krylov and symbolic systems (e.g., Maple's JordanForm/FrobeniusForm) range from $\alpha$4 up to $\alpha$5, with the greatest gains for cases with high block multiplicity and many small Jordan blocks.

A representative trend: for Jordan structure $\alpha$6 and $\alpha$7, total time fell from approximately $\alpha$8 s (naive), $\alpha$9 s (optimized), to $\alpha \in K$00 s (matrix-form), with Maple's time at $\alpha \in K$01 s.

6. Practical Considerations and Open Directions

The algorithm achieves optimal performance when one or more of the following hold: $\alpha \in K$02 (few generalized eigenvectors per factor), or $\alpha \in K$03 has many irreducible factors. Large single Jordan blocks increase Phase III cost but still offer advantages over algebraic extension methods. Growth of coefficients in $\alpha \in K$04 is mitigated by modular preconditioning or early column reductions. Prospective research avenues include multi-prime modular arithmetic, parallelization across ranks, and dynamic adaptation to numeric schemes for small blocks.

This approach is currently most effective for exact arithmetic in $\alpha \in K$05, especially where traditional approaches that require algebraic extensions become computationally prohibitive (Tajima et al., 3 Oct 2025).