The Eigenvector-Eigenvalue Identity and Radar Polarimetry

Eigenvalues and eigenvectors of the polarimetric covariance or coherency matrices play an integral role either directly or indirectly in the analysis and classification of radar polarimetric data. Since these matrices are hermitian their eigenvalues are real and are found simply as the roots of the characteristic polynomial of these matrices. Given an eigenvalue of a matrix a corresponding eigenvector can be found by solving a system of linear equations. There is an interesting and not well known identity, called the eigenvector-eigenvalue identity, that relates the magnitude of the components of an eigenvector to the eigenvalues of the minors of an hermitian matrix. This identity has been rediscovered many times and was recently highlighted in the blog of Fields Medalist Terrance Tao and in a subsequent article. In this paper we illustrate the use of the eigenvector-eigenvalue identity in the context of radar polarimetry using a vegetation model developed by Van Zyl, Arii and Kim. It seems very apropos to dedicate an article emphasizing eigenvectors and eigenvalues in radar polarimetry to the memory of Jakob Van Zyl as these played such a central role in his research.