Recovering Complex Unitary Eigenspaces from Real-Valued Embeddings

TL;DR

This paper presents a method to recover the eigendecomposition of a complex unitary matrix from its real-valued embedding, resolving ambiguities caused by degeneracies and complex conjugates.

Contribution

It introduces a structured projection and orthonormalization procedure that guarantees accurate recovery of eigenvalues and eigenvectors of the original matrix.

Findings

The method resolves eigenspace ambiguities in real-valued embeddings of complex matrices.

It guarantees correct eigenvalue multiplicities and eigenbasis recovery.

The approach is applicable in scientific computing workflows using real-arithmetic solvers.

Abstract

We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ real-arithmetic solvers by representing complex matrices in term of their real and imaginary parts. While the reconstruction is trivial when the spectrum of the real-valued embedding is simple, degenerate and/or complex conjugated eigenvalues introduce ambiguities because each eigenspace may include contributions from both the unitary matrix and its complex conjugate. We prove that this ambiguity can always be resolved by applying a structured projection to the eigenspaces of the real-valued embedding, followed by a rank-revealing orthonormalization. The resulting procedure recovers the eigenvalues and an unitary eigenbasis for the original unitary matrix, with correct multiplicities of degenerate eigenvalues.