Testing the similarity of covariance matrices (CMs) from groups of observations has been shown to be a relevant approach for change and/or anomaly detection in synthetic aperture radar images. Although the term “similarity” usually refers to equality or proportionality, we explore the testing of shared properties in the structure of low rank (LR) plus identity CM, which are appropriate for radar processing. Specifically, we derive two new generalized likelihood ratio tests to infer 1) on the equality of the LR signal component of CMs and 2) on the proportionality of the LR signal component of CMs. The formulation of the second test involves nontrivial optimization problems for which we tailor efficient majorization-minimization algorithms. Eventually, the proposed detection methods enjoy interesting properties that are illustrated on simulations and on an application to real data for change detection.