This paper explores the many-body localization (MBL) transition in a random-field spin-1/2 chain using exact diagonalization. The study reveals that for weak random fields, eigenstates are thermal, indicating an ergodic phase, while for strong fields, eigenstates are localized with short-range entanglement, suggesting a localized phase. The transition occurs at a critical random field strength $ h_c \cong 3.5 \pm 1.0 $. The transition is found to exhibit infinite-randomness scaling with a dynamic critical exponent $ z \rightarrow \infty $, indicating it belongs to an infinite-randomness universality class. The MBL transition is a quantum phase transition at nonzero temperature where equilibrium quantum statistical mechanics breaks down. In the localized phase, the system fails to thermalize, and eigenstates do not thermalize, leading to non-extensive entanglement entropy. In contrast, the ergodic phase allows thermalization, with extensive entanglement entropy. The study examines the behavior of many-body eigenstates in both phases, focusing on local magnetization differences, spin transport, and spin correlations. In the ergodic phase, local magnetization differences decrease exponentially with system size, while in the localized phase, these differences remain large. Spin transport is absent in the localized phase, and spin correlations decay exponentially with distance, unlike the long-range correlations in the ergodic phase. The analysis of energy level statistics shows that in the ergodic phase, the energy spectrum follows Gaussian orthogonal ensemble (GOE) statistics, while in the localized phase, it follows Poisson statistics. The probability distributions of long-distance spin correlations also indicate a transition between these phases, with broad distributions near the critical point. The study also investigates the dynamics of spin modulation, finding that the Thouless energy scales with the many-body level spacing at the transition, suggesting an infinite dynamic critical exponent $ z \rightarrow \infty $. This supports the idea that the MBL transition is governed by an infinite-randomness fixed point. Overall, the results suggest that the MBL transition is a quantum phase transition at nonzero temperature, belonging to an infinite-randomness universality class with a dynamic critical exponent $ z \rightarrow \infty $. The study highlights the importance of exact diagonalization in understanding the properties of many-body eigenstates in both phases of the transition.This paper explores the many-body localization (MBL) transition in a random-field spin-1/2 chain using exact diagonalization. The study reveals that for weak random fields, eigenstates are thermal, indicating an ergodic phase, while for strong fields, eigenstates are localized with short-range entanglement, suggesting a localized phase. The transition occurs at a critical random field strength $ h_c \cong 3.5 \pm 1.0 $. The transition is found to exhibit infinite-randomness scaling with a dynamic critical exponent $ z \rightarrow \infty $, indicating it belongs to an infinite-randomness universality class. The MBL transition is a quantum phase transition at nonzero temperature where equilibrium quantum statistical mechanics breaks down. In the localized phase, the system fails to thermalize, and eigenstates do not thermalize, leading to non-extensive entanglement entropy. In contrast, the ergodic phase allows thermalization, with extensive entanglement entropy. The study examines the behavior of many-body eigenstates in both phases, focusing on local magnetization differences, spin transport, and spin correlations. In the ergodic phase, local magnetization differences decrease exponentially with system size, while in the localized phase, these differences remain large. Spin transport is absent in the localized phase, and spin correlations decay exponentially with distance, unlike the long-range correlations in the ergodic phase. The analysis of energy level statistics shows that in the ergodic phase, the energy spectrum follows Gaussian orthogonal ensemble (GOE) statistics, while in the localized phase, it follows Poisson statistics. The probability distributions of long-distance spin correlations also indicate a transition between these phases, with broad distributions near the critical point. The study also investigates the dynamics of spin modulation, finding that the Thouless energy scales with the many-body level spacing at the transition, suggesting an infinite dynamic critical exponent $ z \rightarrow \infty $. This supports the idea that the MBL transition is governed by an infinite-randomness fixed point. Overall, the results suggest that the MBL transition is a quantum phase transition at nonzero temperature, belonging to an infinite-randomness universality class with a dynamic critical exponent $ z \rightarrow \infty $. The study highlights the importance of exact diagonalization in understanding the properties of many-body eigenstates in both phases of the transition.