This paper presents a new algebraic framework for robust stability analysis of linear time invariant systems with an emphasis on symmetry. The main motivation for this work is to provide a unified theory to answer when the the KYP lemma provides an exact LMI test for robust stability. The notions of weak and strong mutual losslessness are introduced to characterize for lossless S-procedures and the KYP lemma. The new framework has sufficient flexibility to unify some recent extensions of the KYP lemma, including the Generalized KYP lemma for finite frequency analysis, the KYP lemma for nD systems, and the diagonal KYP lemma for positive systems. Finally, we show that the new theory also suggests that the structured singular value of internally positive systems with arbitrary number of scalar uncertainties can be exactly computed.