  The linear quadratic regulator is the fundamental problem of optimal control.
Its state feedback version was set and solved in the early 1960s. However the
static output feedback problem has no explicit-form solution. It is suggested
to look at both of them from another point of view as matrix optimization
problems, where the variable is a feedback matrix gain. The properties of such
a function are investigated, it turns out to be smooth, but not convex, with
possible non-connected domain. Nevertheless, the gradient method for it with
the special step-size choice converges to the optimal solution in the state
feedback case and to a stationary point in the output feedback case. The
results can be extended for the general framework of unconstrained optimization
and for reduced gradient method for minimization with equality-type
constraints.
