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The eigenvector-eigenvalue identity relates the eigenvectors of a Hermitianmatrix to its eigenvalues and the eigenvalues of its principal submatrices inwhich the jth row and column have been removed. We show that one-dimensionalarrays of coupled resonators, described by square matrices with realeigenvalues, provide simple physical systems where this formula can be appliedin practice. The subsystems consist of arrays with the jth resonator removed,and thus can be realized physically. From their spectra alone, the oscillationmodes of the full system can be obtained. This principle of successive singleresonator deletions is demonstrated in two experiments of coupledradiofrequency resonator arrays with greater-than-nearest neighbor couplings,in which the spectra are measured with a network analyzer. Both the Hermitianas well as a non-Hermitian case are covered in the experiments. In both casesthe experimental eigenvector estimates agree well with numerical simulations ifcertain consistency conditions imposed by system symmetries are taken intoaccount. In the Hermitian case, these estimates are obtained from resonancespectra alone without knowledge of the system parameters. It remains aninteresting problem of physical relevance to find conditions under which thefull non-Hermitian eigenvector set can be obtained from the spectra alone.

Recovery of eigenvectors from eigenvalues in systems of coupled harmonic oscillators