On Nonsingularity Verification of Uncertain Matrices Over a Quadratically Constrained Set

Nonsingularity of a matrix is investigated in this technical note that depends affinely on several uncertainty matrices whose real and imaginary parts are constrained by some quadratic forms. Based on some equivalent descriptions for these uncertainty matrices, a sufficient condition is derived. This condition is given in a linear matrix inequality form and can in principle be efficiently validated. The obtained result includes some well known results in systems and control theory as a special case, such as the Lyapunov equation based stability criterion for linear time invariant systems, the (D,G)-scaling based upper bound for the mixed structured singular value, a sufficient condition for the stability of spatially interconnected systems, etc. A computationally verifiable condition is also provided that indicates situations under which the aforementioned condition becomes necessary.