Redundancy in linear optimum regulator problem

The optimum linear regulator problem is the determination of the input to
\dot{x} = Fx + Gu
that minimizes
\int \liminf{0} \limsup{\infty} x^{T}Qx + u^{T}u dt
, and the well-known solution is a feedback law
u = K^{T}x
. It is known that the problem statement is redundant, in that distinct matrices Q
1
and Q
2
can yield the same feedback law
K
. Such matrices are called equivalent, and a simple test for equivalence is available for the single-input case. This note generalizes the test to the multivariable case.