Let K be an arbitrary compact convex set of square matrices, This paper presents necessary and sufficient conditions for the spectrum of K to have no eigenvalues in a prespecified closed con,ex subset of the complex plane, The obtained result implies different criteria for analysis of the spectral set of K. In particular, we have formulated criteria for nonsingularity, inertia and Hurwitz stability of K which can be used in the robustness analysis of linear control systems with uncertain parameters.