This paper introduces several stability conditions for a given class of matrices expressed in terms of Linear Matrix Inequalities (LMI), being thus simply and efficiently computable. Diagonal and simultaneous stability, both characterized by polytopes of matrices, are addressed. Using this approach a method particularly attractive to test a given matrix for D-stability is proposed. Lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions. The key idea is to relate Hurwitz stability with a positive realness condition. © 1998 Elsevier Science Inc. All rights reserved.2851-36980Boyd, S.P., Ghaoui, L.E., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , SIAM, PhiladelphiaColaneri, P., Geromel, J.C., Locatelli, A., (1997) Control Theory and Design: An RH2 and RH∞ Viewpoint, , Academic Press, New YorkFeron, E., Apkarian, P., Gahinet, P., Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions (1996) IEEE Trans. Aut. Contr., 41 (7), pp. 1041-1046Geromel, J.C., On the determination of a diagonal solution of the Lyapunov equation (1985) IEEE Trans. Aut. Contr., 30 (4), pp. 404-406Geromel, J.C., Peres, P.L.D., Bernussou, J., On a convex parameter space method for linear control design of uncertain systems (1991) SIAM Journal on Control and Optimization, 29 (2), pp. 381-402Hershkowitz, D., Recent directions in matrix stability (1992) Linear Algebra Appl., 171, pp. 161-186Kaszkurewicz, E., Bhaya, A., Robust stability and diagonal Lyapunov functions (1993) SIAM J. Matrix Anal. Appl., 14 (2), pp. 508-520Sun, W., Khargonekar, P.P., Shim, D., Solution to the positive real control problem for linear time invariant systems (1994) IEEE Trans. Aut. Contr., 39 (10), pp. 2034-2046Persidskii, S.K., Problem on absolute stability (1969) Automat. Remote Control, 12, pp. 1889-1895