Invariance of the distributions of normalized Gram matrices

Normalized Gram matrices formed from multiple vectors of sensor data, and functions of the eigenvalues of such matrices in particular, have a long history in connection with multiple-channel detection. The determinant and various other functions of the eigenvalues of these matrices arise as detection statistics in a variety of multichannel problems, and knowledge of their distributions under the H₀ assumption that the sensor channels are independent and contain only white gaussian noise is consequently important for determining false-alarm probabilities for multi-channel detectors. Invariance of the H₀ distribution of the eigenvalues to one data channel is significant in some applications. This paper derives the H₀ distribution of a normalized Gram matrix and, as corollaries, obtains the distribution of the determinant as well as invariance results for the matrix that carry over to its spectrum. The essential symmetry property of white gaussian noise on which these results depend is also noted.