This paper presents a general class of problems for detecting subspace signals in subspace interference and broadband noise. The authors derive the generalized likelihood ratio (GLR) for each problem in the class and establish its invariances, showing that these are the natural invariances for the problem. They argue that the GLR test (GLRT) is the uniformly most powerful invariant detector. The GLRT is shown to be optimal for all finite sample sizes, improving on the standard asymptotic theory of the GLRT. The paper discusses detection problems involving unknown parameters in the mean and covariance of a multivariate normal (MVN) distribution. The detection problem is described as a test of two hypotheses: the null hypothesis that the data consists of noise only, and the alternative hypothesis that the data consists of a signal and noise. The signal is assumed to lie in a subspace, and the noise is assumed to be MVN with known or unknown variance. The paper introduces the concept of invariance in the context of GLRTs and shows that the GLRT is invariant to a natural set of invariances. This invariance is crucial for the optimality of the GLRT. The authors also show that the GLRT is a monotone function of the uniformly most powerful invariant (UMP-invariant) tests derived in previous work. The paper provides several examples of detection problems, including detection in unknown bias and detection in sinusoidal interference. The authors also discuss the linear algebraic preliminaries necessary for the analysis of the GLRT, including the projection operators and their properties. The paper concludes by showing that the GLRT is UMP-invariant for testing the hypotheses in the detection problems discussed. The false alarm and detection probabilities are derived for the GLRT, and the results are illustrated with ROC curves and detector diagrams. The paper generalizes previous results and provides a comprehensive analysis of the GLRT for subspace signal detection in subspace interference and broadband noise.This paper presents a general class of problems for detecting subspace signals in subspace interference and broadband noise. The authors derive the generalized likelihood ratio (GLR) for each problem in the class and establish its invariances, showing that these are the natural invariances for the problem. They argue that the GLR test (GLRT) is the uniformly most powerful invariant detector. The GLRT is shown to be optimal for all finite sample sizes, improving on the standard asymptotic theory of the GLRT. The paper discusses detection problems involving unknown parameters in the mean and covariance of a multivariate normal (MVN) distribution. The detection problem is described as a test of two hypotheses: the null hypothesis that the data consists of noise only, and the alternative hypothesis that the data consists of a signal and noise. The signal is assumed to lie in a subspace, and the noise is assumed to be MVN with known or unknown variance. The paper introduces the concept of invariance in the context of GLRTs and shows that the GLRT is invariant to a natural set of invariances. This invariance is crucial for the optimality of the GLRT. The authors also show that the GLRT is a monotone function of the uniformly most powerful invariant (UMP-invariant) tests derived in previous work. The paper provides several examples of detection problems, including detection in unknown bias and detection in sinusoidal interference. The authors also discuss the linear algebraic preliminaries necessary for the analysis of the GLRT, including the projection operators and their properties. The paper concludes by showing that the GLRT is UMP-invariant for testing the hypotheses in the detection problems discussed. The false alarm and detection probabilities are derived for the GLRT, and the results are illustrated with ROC curves and detector diagrams. The paper generalizes previous results and provides a comprehensive analysis of the GLRT for subspace signal detection in subspace interference and broadband noise.