An elementary proof for the exactness of (D,G) scaling

The goal of this paper is to provide an elementary proof for the exactness of the (D,G) scaling applied to the uncertainty structure with one repeated real scalar block and one full complex matrix block. The (D,G) scaling has vast application area around control theory, optimization and signal processing. This is because, by applying the (D,G) scaling, we can convert inequality conditions depending on an uncertain parameter to linear matrix inequalities (LMIs) in an exact fashion. However, its exactness proof is tough, and this stems from the fact that the proof requires an involved matrix formula in addition to the standard Lagrange duality theory. To streamline the proof, in the present paper, we clarify that the involved matrix formula is closely related to a norm preserving dilation under structural constraints. By providing an elementary proof for the norm preserving dilation, it follows that basic results such as Schur complement and congruence transformation in conjunction with the Lagrange duality theory are enough to complete a self-contained exactness proof.