This paper considers linear precoder optimization problems for multiple-input multiple-output (MIMO) systems. In addition to the conventionally used sum-power constraint, maximum eigenvalue constraint on the precoding matrix is also considered so as to account for power limitations imposed on each antenna by the linearity of its own power amplifier in practical implementations. A framework employing directional derivative is developed to obtain optimal precoder designs for different criteria including maximizing the information rate and minimizing the sum of mean-square error (MSE). It turns out that power allocations in such situations are piecewise linear in sum-power space. The piecewise linear property allows us to generate the entire path of solution through finding out a finite number of breakpoints. A Homotopy-type algorithm is then proposed to obtain the solution for an arbitrary sum-power constraint. The number of breakpoints to be determined in our exact piecewise linear solution is in fact only about two times of the number of transmit antennas, so that our method is super fast and outperforms existing approximate solutions in the literature in both effectiveness and efficiency. Simulated experiments are performed to verify our theoretical analysis.