  Systems with an effectively non-Hermitian Hamiltonian display an enhanced
sensitivity to parametric and dynamic perturbations, which arises from the
nonorthogonality of their eigenstates. This enhanced sensitivity can be
quantified by the phase rigidity, which mathematically corresponds to the
eigenvalue condition number, and physically also determines the Petermann
factor of quantum noise theory. I derive an exact nonperturbative expression
for this sensitivity measure that applies to arbitrary eigenvalue
configurations. The expression separates spectral correlations from additional
geometric data, and retains a simple asymptotic behaviour close to exceptional
points (EPs) of any order, while capturing the role of additional states in the
system. This reveals that such states can have a sizable effect even if they
are spectrally well separated, and identifies the specific matrix whose
elements determine this nonperturbative effect. The employed algebraic
approach, which follows the eigenvectors-from-eigenvalues school of thought,
also provides direct insights into the geometry of the states near an EP. For
instance, it can be used to show that the phase rigidity follows a striking
equipartition principle in the quasi-degenerate subspace of a system.
