This paper explores a toy model for the study of monitored dynamics in many-body quantum systems, focusing on the stochastic Schrödinger equation resulting from continuous monitoring of a random Hermitian operator drawn from the Gaussian Unitary Ensemble (GUE) at each time \( t \). The dynamics of the eigenvalues of the density matrix decouple from the eigenvectors due to unitary invariance, leading to stochastic equations that describe the evolution of the eigenvalues. The study considers two regimes: one with an additional dephasing term due to imperfect quantum measurements, where the density matrix has a stationary distribution described by the inverse-Marchenko-Pastur distribution in the limit of large \( n \); and another with perfect measurements, where purification occurs, and the focus is on finite-time dynamics. An exact solution for the joint probability distribution of the eigenvalues at each time \( t \) and for each size \( n \) is derived, revealing two regimes: a Coulomb gas regime at short times \( t \Gamma = O(1) \) and a regime with exponentially separated eigenvalues at large times \( t \Gamma = O(n) \). The paper also analyzes the asymptotic behaviors of the entanglement entropy in these regimes.This paper explores a toy model for the study of monitored dynamics in many-body quantum systems, focusing on the stochastic Schrödinger equation resulting from continuous monitoring of a random Hermitian operator drawn from the Gaussian Unitary Ensemble (GUE) at each time \( t \). The dynamics of the eigenvalues of the density matrix decouple from the eigenvectors due to unitary invariance, leading to stochastic equations that describe the evolution of the eigenvalues. The study considers two regimes: one with an additional dephasing term due to imperfect quantum measurements, where the density matrix has a stationary distribution described by the inverse-Marchenko-Pastur distribution in the limit of large \( n \); and another with perfect measurements, where purification occurs, and the focus is on finite-time dynamics. An exact solution for the joint probability distribution of the eigenvalues at each time \( t \) and for each size \( n \) is derived, revealing two regimes: a Coulomb gas regime at short times \( t \Gamma = O(1) \) and a regime with exponentially separated eigenvalues at large times \( t \Gamma = O(n) \). The paper also analyzes the asymptotic behaviors of the entanglement entropy in these regimes.