A General Test for the Linear Structure of Covariance Matrices of Gaussian Populations

This paper addresses the problem of testing whether a covariance matrix can be expressed by an unknown linear combination of a set of known matrices or by another unknown linear combination of a set of different, but known, matrices. This problem is of interest in a wide range of real-world applications, such as radar, sonar, and spectrum sensing. We study the problem under the Gaussian assumption and derive the generalized likelihood ratio test (GLRT). Since there is no general closed-form solution for the maximum likelihood (ML) estimates of the covariance matrices, which are required for the GLRT, we resort to a powerful inverse iteration algorithm. Finally, an example, along with numerical results, is given to illustrate the methodology.