EEG-based brain fingerprint identification usually requires participators to complete a particular task under external stimuli, such as recognition based on visual-evoked potentials (VEP) and resting potential (RP). These paradigms often fail to generalize due to their dependency on task-specific brain responses, which restricts their applicability in dynamic real-world environments. This chapter proposes a fast task-free brain fingerprint identification method based on Low-Rank and Matrix Decomposition (LRMD). Task-related EEG signals are conceptually divided into two components: the background EEG (BEEG) and the residue EEG (REEG). The BEEG encapsulates an individual’s intrinsic and unique brain print features that remain stable over time. Moreover, only a subset of these brainprint features is sufficient to characterize an individual’s identity. Consequently, the BEEG can be represented by a compact set of basis vectors in its feature space, exhibiting low-rank characteristics. The LRMD framework effectively isolates identity-related components by decomposing raw EEG signals into meaningful subspaces while filtering out noise and irrelevant features. This capability allows for accurate recognition even without requiring participants to engage in predefined tasks or external stimuli. Extensive experiments are conducted on three public EEG datasets and a self-collected multi-task EEG dataset, demonstrating outstanding performance across varying low-rank configurations and diverse time scales. These results indicate that the proposed method is robust and task-agnostic, further validating its potential for practical applications. The best results can reach accuracy above 99.90%, highlighting its potential for widespread application in practical scenarios such as authentication and secure access systems.