Abstract: Given many copies of an unknown quantum state "rho", we introduce the task of learning a classical description of its principal eigenstate. Namely, assuming that rho has an eigenstate |phi> with eigenvalue lambda > 1/2, the goal is to learn a classical description of |phi> which can later be used to estimate expectation values <phi|O|phi> for any O in some class of observables. We argue that learning the principal eigenstate rather than the state itself is the more natural setting for many applications of classical shadows, and in fact, can also lead to a reduction in sample complexity. We give a sample-efficient algorithm for this task that compares favorably to the naive approach of applying quantum state purification followed by the single-copy classical shadows scheme. In fact, in the regime where lambda is sufficiently large, the sample complexity of our algorithm matches the optimal sample complexity for pure state classical shadows (i.e., when lambda=1 exactly).