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An Algorithm for Diagonalizing Matrices of Formal Power Series

Zihao Dai, Hao Liang, Jingyu Lu, Lihong Zhi

TL;DR

The paper extends the classical spectral theorem to matrices over multiparameter formal power series by establishing a necessary-and-sufficient condition for unitary diagonalization: a normal matrix is unitarily diagonalizable precisely when its minimal polynomial splits completely over the ring and the spectral projections lie in the ring. It then delivers a constructive algorithm to decide diagonalizability over regular local rings, hinging on a Jacobian-based test for complete splitting that avoids full factorization in power-series rings. By introducing a ramification-theoretic, Henselian framework and the universal decomposition algebra, the authors connect root behavior in formal completions to explicit algebraic criteria, enabling the computation of unitary diagonalizations via spectral projections and Gram–Schmidt. Although the integral-closure step is computationally heavy, the work outlines clear directions to overcome this bottleneck, including leveraging Galois symmetries and hybrid analytic-algebraic techniques for practical perturbation problems. The results hold potential for multiparameter perturbation theory and algebraic-geometry–informed diagonalization of parameter-dependent operators.

Abstract

This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the associated spectral projections have entries in the ring. Building on this characterization, we develop an algorithm for deciding the unitary diagonalizability of matrices over regular local rings of algebraic varieties. A central ingredient of the algorithm is a decision procedure for determining whether a polynomial splits over a formal power series ring; we establish this using techniques from prime decomposition and the relative smoothness of integral closures in ramification theory.

An Algorithm for Diagonalizing Matrices of Formal Power Series

TL;DR

The paper extends the classical spectral theorem to matrices over multiparameter formal power series by establishing a necessary-and-sufficient condition for unitary diagonalization: a normal matrix is unitarily diagonalizable precisely when its minimal polynomial splits completely over the ring and the spectral projections lie in the ring. It then delivers a constructive algorithm to decide diagonalizability over regular local rings, hinging on a Jacobian-based test for complete splitting that avoids full factorization in power-series rings. By introducing a ramification-theoretic, Henselian framework and the universal decomposition algebra, the authors connect root behavior in formal completions to explicit algebraic criteria, enabling the computation of unitary diagonalizations via spectral projections and Gram–Schmidt. Although the integral-closure step is computationally heavy, the work outlines clear directions to overcome this bottleneck, including leveraging Galois symmetries and hybrid analytic-algebraic techniques for practical perturbation problems. The results hold potential for multiparameter perturbation theory and algebraic-geometry–informed diagonalization of parameter-dependent operators.

Abstract

This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the associated spectral projections have entries in the ring. Building on this characterization, we develop an algorithm for deciding the unitary diagonalizability of matrices over regular local rings of algebraic varieties. A central ingredient of the algorithm is a decision procedure for determining whether a polynomial splits over a formal power series ring; we establish this using techniques from prime decomposition and the relative smoothness of integral closures in ramification theory.
Paper Structure (5 sections, 6 theorems, 61 equations, 2 algorithms)

This paper contains 5 sections, 6 theorems, 61 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Criterion for Unitary Diagonalization over $\mathbf{C}$
  3. When are the eigenvalues power series?
  4. When are Spectral Projections in $\operatorname{Mat}_n(\mathbf{C})$?
  5. Conclusions and Future Work

Key Result

Theorem 1

A normal matrix $A$ is diagonalizable under the adjoint action of the unitary group $\mathbb{U}_n(\mathbf{C})$ if and only if the characteristic polynomial of $A$ totally splits over $\mathbf{C}$ and all the matrix entries of the projection operators onto the eigen-spaces of $A$ belong to $\mathbf{C and $A = \sum_k\lambda_k \Pi_{k}$.

Theorems & Definitions (23)

  • Example 1
  • Theorem 1
  • proof
  • Remark 1
  • Example 2
  • Example 3
  • Proposition 1
  • proof
  • Theorem 2
  • Lemma 1
  • ...and 13 more