A cooperative blind spectrum detection with the Doolittle decomposition and support vector machine (SVM) classification is proposed to improve the performance and reduce the complexity and latency of spectrum detection in hybrid Geostationary Earth Orbit and Low Earth Orbit (GEO-LEO) satellite constellation networks. First, a principal component analysis (PCA) algorithm is adopted to process perceive signals. Second, the maximum-to-minimum eigenvalue ratio of a covariance matrix is utilized to construct statistics with self-learning ability by using the Doolittle decomposition. Third, primary user and noise signals are labeled with “+1” and “$-$1” as training labels, respectively. Finally, a high-performance PCA-SVM classification is generated by training the labels and the corresponding statistics. Innovations mainly include the Doolittle decomposition of the covariance matrix in the SVM and the PCA to preprocess the perceived signals. Simulation results show that the detection probability of the proposed scheme outperforms the energy detection and the SVM algorithm by 55 and 15% at a signal-to-noise ratio (SNR) of $-$15 dB, respectively. The average error probability of the proposed scheme is 20% lower than that of the SVM at an SNR of $-$20 dB. Therefore, in hybrid GEO-LEO satellite constellation networks, the cooperative blind spectrum detection with the Doolittle decomposition and SVM classification can detect the spectrum with low complexity and high performance effectively.