  The eigenvector-eigenvalue identity relates the eigenvectors of a Hermitian
matrix to its eigenvalues and the eigenvalues of its principal submatrices in
which the jth row and column have been removed. We show that one-dimensional
arrays of coupled resonators, described by square matrices with real
eigenvalues, provide simple physical systems where this formula can be applied
in practice. The subsystems consist of arrays with the jth resonator removed,
and thus can be realized physically. From their spectra alone, the oscillation
modes of the full system can be obtained. This principle of successive single
resonator deletions is demonstrated in two experiments of coupled
radiofrequency resonator arrays with greater-than-nearest neighbor couplings,
in which the spectra are measured with a network analyzer. Both the Hermitian
as well as a non-Hermitian case are covered in the experiments. In both cases
the experimental eigenvector estimates agree well with numerical simulations if
certain consistency conditions imposed by system symmetries are taken into
account. In the Hermitian case, these estimates are obtained from resonance
spectra alone without knowledge of the system parameters. It remains an
interesting problem of physical relevance to find conditions under which the
full non-Hermitian eigenvector set can be obtained from the spectra alone.
