Is it true that the state-space realization above is minimal if and only if there exists a real $\alpha$ for which the following two matrix equations have symmetric and positive definite solutions P and W? Prove or give counter example.
$A^{T}P + PA + \alpha P= -BB^{T}$
$AW + WA^{T} + \alpha W= -C^{T}C$
I know that a system is minimal if and only if it is controllable and observable. My first instinct was to use the PBH test for controllability and observability but the form of the equations does not lend itself to that test. Thoughts?