An Algorithm for Diagonalizing Matrices of Formal Power Series

TL;DR

This paper presents a characterization and algorithm for determining when matrices over formal power series rings are unitarily diagonalizable, based on their minimal polynomial and spectral projections, with applications to algebraic varieties.

Contribution

It provides a new criterion for diagonalizability of matrices over formal power series rings and develops an algorithm leveraging prime decomposition and ramification theory.

Findings

Normal matrices are unitarily diagonalizable iff their minimal polynomial splits and spectral projections are in the ring.

An algorithm is developed for deciding diagonalizability over regular local rings.

The algorithm uses polynomial splitting and techniques from algebraic geometry.

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.