Recovering Complex Unitary Eigenspaces from Real-Valued Embeddings
It solves a practical ambiguity problem for scientific computing workflows that use real-arithmetic solvers on complex matrices.
The paper proves that a unitary eigendecomposition of a complex unitary matrix can be recovered from its real-valued embedding, even with degenerate or complex conjugated eigenvalues, by applying a structured projection and rank-revealing orthonormalization.
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.