  A Lagrangian formulation for the constrained search for the $N$-representable
one-particle density matrix based on the McWeeny idempotency error minimization
is proposed, which converges systematically to the ground state. A closed form
of the canonical purification is derived for which no a posteriori adjustement
on the trace of the density matrix is needed. The relationship with comparable
methods are discussed, showing their possible generalization through the
hole-particle duality. The appealing simplicity of this self-consistent
recursion relation along with its low computational complexity could prove
useful as an alternative to diagonalization in solving dense and sparse matrix
eigenvalue problems.
