Symbolic spectral decomposition of 3x3 matrices

TL;DR

This paper presents a symbolic spectral decomposition method for 3x3 matrices that improves numerical accuracy over existing approaches, especially in limit cases, by reformulating matrix invariants as sum-of-products expressions.

Contribution

It introduces an alternative symbolic approach for computing matrix invariants that enhances floating point accuracy in spectral decomposition of 3x3 matrices.

Findings

Increased numerical accuracy in spectral decomposition, especially in eigenvalue multiplicity cases.

Demonstrated robustness of the method through numerical examples.

Potential for integration into engineering applications requiring precise spectral analysis.

Abstract

Spectral decomposition of matrices is a recurring and important task in applied mathematics, physics and engineering. Many application problems require the consideration of matrices of size three with spectral decomposition over the real numbers. If the functional dependence of the spectral decomposition on the matrix elements has to be preserved, then closed-form solution approaches must be considered. Existing closed-form expressions are based on the use of principal matrix invariants which suffer from a number of deficiencies when evaluated in the framework of finite precision arithmetic. This paper introduces an alternative form for the computation of the involved matrix invariants (in particular the discriminant) in terms of sum-of-products expressions as function of the matrix elements. We prove and demonstrate by numerical examples that this alternative approach leads to increased floating point accuracy, especially in all important limit cases (e.g. eigenvalue multiplicity). It is believed that the combination of symbolic algorithms with the accuracy improvements presented in this paper can serve as a powerful building block for many engineering tasks.