This paper formulates 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 and establish its invariances, arguing that these invariances are natural and optimal. The GLR is shown to be a maximal invariant statistic, and its distribution is monotone, making the GLRT (Generalized Likelihood Ratio Test) the uniformly most powerful invariant detector. The paper illustrates the utility of the GLRT by solving several problems, including detecting subspace signals in subspace interference and broadband noise, and provides the distribution of the detector and performance curves. The GLRT is proven to be optimal for all finite sample sizes, improving on standard asymptotic theory. The paper also discusses the optimality and performance of the GLRT, providing ROC curves and detector diagrams for various scenarios.This paper formulates 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 and establish its invariances, arguing that these invariances are natural and optimal. The GLR is shown to be a maximal invariant statistic, and its distribution is monotone, making the GLRT (Generalized Likelihood Ratio Test) the uniformly most powerful invariant detector. The paper illustrates the utility of the GLRT by solving several problems, including detecting subspace signals in subspace interference and broadband noise, and provides the distribution of the detector and performance curves. The GLRT is proven to be optimal for all finite sample sizes, improving on standard asymptotic theory. The paper also discusses the optimality and performance of the GLRT, providing ROC curves and detector diagrams for various scenarios.