Recovery of eigenvectors from eigenvalues in systems of coupled harmonic oscillators

Abstract

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 radio-frequency 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.