The purpose of this short paper is to round out recent results in the literature on fixing the state-feedback gain by choice of closed-loop eigensystem. The constraints on attainable eigensystems and a formula for the feedback gain are derived in a direct manner, and interpreted using the idea of 'eigen-inputs'. The well-known formula for the feedback gain of an optimal quadratic regulator is re-examined in this light; using a polynomial-matrix-fraction, a matrix generalization of the scalar 'root-square characteristic equation' is also obtained.