Conditions for the Existence of a Positive Definite Solution to the Linear Modified Lyapunov Equation

A new sufficient condition for the existence of a positive definite solution to the linear modified Lyapunov equation is derived. With an additional mild assumption this condition is also proven necessary. Based on this result, the largest size of uncertainty for which a given feedback leads to robust stabilization based on the linear modified Lyapunov equation is expressed in closed form. This radius is proven equal to the reciprocal of the largest positive real eigenvalue of a specific "guarding matrix".