This article aims at a distributed design to minimize the nuclear norm (the sum of all singular values) under linear equality constraints over a multiagent network. The problem is reformulated as a distributed trace norm minimization problem by introducing substitutional variables. A distributed projected primal-dual algorithm is proposed for the reformulation. It is shown that the algorithm converges to an optimal solution with a rate of O(1/t). Numerical simulations on three classical problems, including linear matrix equality constraints, cardinality minimization, and low-rank matrix completion, are carried out for illustration.