Generalizations of Some Criteria for P-Matrices and M-Matrices
نویسنده
چکیده
It is well known that the set of P-matrices includes several important classes of matrices such as M-matrices and positive definite matrices. P-matrices arize in various theoretical and applied fields, for example in linear complementarity theory [9], in the analysis of the solution set of systems of linear interval equations [6], in the study of convex sets of matrices [5]. The class of M-matirces is characterized by a special sign pattern of matrix elements which suggests relations with the theory of nonnegative matrices. From an application point of view, M-matrices play an important role in certain economic models [8] and provide a tool for stability analysis of composite dynamical systems [1]. There is a large number of different in form but essentially equivalent conditions that are necessary and sufficient for a given matrix to be a P-matrix or an M-matrix. Selected lists of such conditions together with the relevant theory are given in [4]. Some generalizations of the P-matrix concept related mainly to the linear complementarity problems can be found in [3], [9] and their references. The results in [7] present criteria for the P-property of all matrices belonging to an interval matrix set and introduce the notion of interval P-matrices. In this paper, we study the P-property and M-property of matrices which are elements of compact and convex matrix sets. Our aim is to establish criteria characterizing these properties with respect to all elements of the matrix set. In Theorems 2.1 and 3.1, we have obtained general criteria which are valid for any compact convex set of matrices. Several special cases of compact convex sets found in the literature are also considered. In each case, in addition to the general criterion, we have derived equivalent necessary and sufficient conditions which provide a finite test for the P-property and M-property of all matrices belonging to the matrix set. The obtained results are based on well known criteria for a single P-matrix and M-matrix and generalize these criteria to the case of compact and convex matrix sets. БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ . BULGARIAN ACADEMY OF SCIENCES
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