Lower bounds to the ground-state expectation value of non-negative operators

A lower bound formula for the ground-state expectation value of a non-negative operator using an approximating trial function is generalized and presented as a matrix-eigenvalue problem which reverts to the original formula in the scalar case. The lower bound is applied to the hydrogen atom and the helium atom to show that significant improvement results from the generalization when the trial function is not of high quality. But as the quality of the trial function increases, data suggest the lower bounds from the generalized method converge to those of the original scalar formula.