[s-kuberski]

This extends the existing GEVP functionality to complex Hermitian correlation matrices. With vector_obs=True one can also propagates the uncertainties of eigenvectors.

For $A=C(t)$ and $B=C(t_0)$, we solve

$$ Av_n=\lambda_n Bv_n,\qquad v_m^\dagger Bv_n=\delta_{mn}, $$

with positive-definite $B$. Error propagation uses the derivatives

$$ \begin{aligned} d\lambda_n &= v_n^\dagger(dA-\lambda_n dB)v_n,\\ dv_n &= \sum_{m\ne n}v_m \frac{v_m^\dagger(dA-\lambda_n dB)v_n}{\lambda_n-\lambda_m} -\frac12v_n(v_n^\dagger dBv_n). \end{aligned} $$

The crucial piece for the error propagation of the eigenvectors is the choice of the phase convention is $\mathrm{Im}(v_n^\dagger B,dv_n)=0$.

These derivatives are passed to derived_observable as a real-coordinate Jacobian, including uncertainties in both matrices and their correlations. Working directly with the complex eigensystem avoids the doubled eigenvalues of the real block representation discussed in #255. Genuine degeneracies still prevent the determination of the derivatives of the eigenvectors.

The existing real-matrix GEVP options are supported for complex input. Complex vector uncertainties use the derivatives above rather than differentiating the Cholesky decomposition. Projections use $v_l^\dagger C v_r$ (before we did not have to worry about the exact definition).

The many tests include matrices with prescribed eigenvalues and eigenvectors, checking their values, errors and correlations.

For now, this draft is intended for testing on the matrices motivating #255.

[Copilot]

🟡 Changes recommended

The real and imaginary component checks mishandle fixed complex entries in matrices containing CObs, leaving imaginary values unchanged in corr.real and returning zeros in corr.imag.

1 open issue

• Handle numeric complex entries in corr.real and corr.imag · New

What changed in this PR

Extends the existing generalized eigenvalue routines to complex Hermitian correlation matrices, including optional propagation of eigenvector uncertainties.

Changes:

• Adds complex eigensystem derivatives with a defined local phase convention.
• Updates projections, pruning, and matrix helpers for complex input, and clarifies solver-failure handling.
• Adds tests for covariance propagation, normalization, sorting, and prescribed eigensystems.

File	Description
tests/​linalg_test.py	Tests complex eigenvector values and propagated covariance.
tests/​correlators_test.py	Tests complex GEVP modes, projections, uncertainties, and failures.
pyerrors/​linalg.py	Implements complex Hermitian eigensystem error propagation.
pyerrors/​correlators.py	Integrates complex GEVP support and conjugated projections.

_{Review effort: Balanced}

[Copilot]

🔵 Needs a closer look

Active issues can discard complex eigenvector components during sorting and raise AttributeError for mixed Obs/complex64 matrices, breaking supported GEVP and eigensolver inputs.

No open issues 🎉

2 previously missed issues

In code that hasn't changed since last review

Promote eigenvector sorting reference to complex

pyerrors/​correlators.py:1503

When C(t0) and C(ts) are real but another timeslice is complex, the sorting reference has a float dtype, so replacing its rows with complex vectors discards their imaginary components. This corrupts the determinant scores and can leave best_perm unassigned, causing GEVP(sort='Eigenvector') to fail. Promote the sorting reference to complex before replacing its rows.

Convert complex64 coordinates before derived-observable processing

pyerrors/​linalg.py:418

A Hermitian object matrix mixing Obs entries with np.complex64 constants leaves np.float32 real and imaginary coordinates here. derived_observable only wraps Python int/float constants before accessing .cov_names, so eigh, eigv, and complex GEVPs with vector_obs=True raise AttributeError for this input. Preserve Obs components and convert the remaining coordinates to Python floats.

1 resolved since last review

• Handle numeric complex entries in corr.real and corr.imag

_{Review effort: Balanced}