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Existence and Enumeration of Polynomially Transformed Matrices under Spectral and Nilpotent Constraints

Shih-Yu Chang

TL;DR

This work investigates how polynomial transforms $f(X)$ interact with the spectral and nilpotent parts of a matrix under Jordan decomposition. It develops a unified framework to address when $f(X)$ is nilpotent or diagonalizable, distinguishing between one- and multiple-eigenvalue structures and between sparse and dense polynomial coefficients. For sparse polynomials, the analysis hinges on root-counting tools (Descartes’ rule, Sturm’s theorem) and combinatorial Jordan-block enumerations; for dense polynomials, it leverages complex-analytic methods (the Argument Principle, Rouché’s theorem) and algebraic techniques (gcd, square-free factorization) to count roots and structure. The results yield explicit counting formulas for similarity classes of $X$ satisfying the nilpotency or diagonalizability constraints and illustrate the relative computational implications of sparse versus dense representations, advancing both operator theory and computational linear algebra.

Abstract

Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan form--capture both diagonalizable and defective matrices, revealing insights into dynamics, stability, and modal interactions. Building on this foundation, this study examines how matrix functions behave under nilpotency and diagonalizability constraints, inspired by the Jordan decomposition into spectral and nilpotent parts. The spectral part encodes eigenstructure, while the nilpotent part reflects deviations from diagonalizability. Two main problems are addressed: (1) identifying when a polynomial-transformed matrix becomes nilpotent, and (2) determining when it becomes diagonalizable. The analysis considers two scenarios--matrices with one distinct eigenvalue and those with multiple distinct eigenvalues--each posing unique algebraic and combinatorial challenges. Depending on whether the polynomial transform is sparse or dense, distinct tools are applied: combinatorial root-counting techniques such as Descartes' rule and Sturm's theorem for sparse cases, and analytic or algebraic methods like the argument principle, Rouche's theorem, and square-free factorization for dense ones. Together, these methods establish a systematic framework for studying the existence and classification of matrices with specific structural properties under polynomial transformations. The findings deepen understanding of spectral transformations and provide new perspectives for operator theory, dynamical systems, and computational linear algebra.

Existence and Enumeration of Polynomially Transformed Matrices under Spectral and Nilpotent Constraints

TL;DR

This work investigates how polynomial transforms interact with the spectral and nilpotent parts of a matrix under Jordan decomposition. It develops a unified framework to address when is nilpotent or diagonalizable, distinguishing between one- and multiple-eigenvalue structures and between sparse and dense polynomial coefficients. For sparse polynomials, the analysis hinges on root-counting tools (Descartes’ rule, Sturm’s theorem) and combinatorial Jordan-block enumerations; for dense polynomials, it leverages complex-analytic methods (the Argument Principle, Rouché’s theorem) and algebraic techniques (gcd, square-free factorization) to count roots and structure. The results yield explicit counting formulas for similarity classes of satisfying the nilpotency or diagonalizability constraints and illustrate the relative computational implications of sparse versus dense representations, advancing both operator theory and computational linear algebra.

Abstract

Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan form--capture both diagonalizable and defective matrices, revealing insights into dynamics, stability, and modal interactions. Building on this foundation, this study examines how matrix functions behave under nilpotency and diagonalizability constraints, inspired by the Jordan decomposition into spectral and nilpotent parts. The spectral part encodes eigenstructure, while the nilpotent part reflects deviations from diagonalizability. Two main problems are addressed: (1) identifying when a polynomial-transformed matrix becomes nilpotent, and (2) determining when it becomes diagonalizable. The analysis considers two scenarios--matrices with one distinct eigenvalue and those with multiple distinct eigenvalues--each posing unique algebraic and combinatorial challenges. Depending on whether the polynomial transform is sparse or dense, distinct tools are applied: combinatorial root-counting techniques such as Descartes' rule and Sturm's theorem for sparse cases, and analytic or algebraic methods like the argument principle, Rouche's theorem, and square-free factorization for dense ones. Together, these methods establish a systematic framework for studying the existence and classification of matrices with specific structural properties under polynomial transformations. The findings deepen understanding of spectral transformations and provide new perspectives for operator theory, dynamical systems, and computational linear algebra.
Paper Structure (39 sections, 12 theorems, 83 equations, 1 table, 1 algorithm)

This paper contains 39 sections, 12 theorems, 83 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Function Transform Review and Problem Formulation
  3. Nilpotency Constraints
  4. One Distinct Eigenvalue
  5. Sparse Coefficients of $f(x)$
  6. Eigenvalues Existence and Counting
  7. Sturm’s Theorem and Exact Root Counts for Sparse Polynomials
  8. Sturm’s Sequence and Sturm’s Theorem sturmfels2002solving
  9. Counting Jordan Block Structures for a Single Eigenvalue
  10. Dense Coefficients of $f(x)$
  11. Existence of Roots
  12. 1. Argument Principle for Root Counting
  13. 2. Rouché’s Theorem for Existence
  14. 3. Comparison of Methods
  15. Multiple Distinct Eigenvalues
  16. ...and 24 more sections

Key Result

Theorem 1

Given an analytic function $f(x)$ within the domain for $|z| < R$, a matrix $\bm{X}$ with the dimension $m$ and $K$ distinct eigenvalues $\lambda_k$ for $k=1,2,\ldots,K$ such that where $\left\vert\lambda_k\right\vert<R$, and matrices $\bm{P}_{k,i}$ and $\bm{N}_{k,i}$ are projector and nilpotent matrices with respect to the $k$-th eigenvalue and its $i$-th geometry component. Then, we have

Theorems & Definitions (32)

  • Theorem 1
  • Example 1
  • Example 2
  • Theorem 2: Distinct-root count via gcd with derivative
  • proof
  • Corollary 1: Simple-root criterion / separability
  • proof
  • Theorem 3: Square--free factorization and number of distinct roots
  • proof
  • Remark 1
  • ...and 22 more