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For n= 2, 3, and 4, conditions are given for the real n by n D-stable matri ces. The 3 by 3 sufficient condition is easily checkable and reveals to be D-stable a class of matrices which is not included in any known, ge neral sufficient condition. The concept of D-stability was originally introduced in the economic literature by Arrow and McManus [1] I with a stronger definition_ We shall adopt the definition of fairly common current usage_ Let M n (R) de note th e class of n by n matrices over the real field and denote by a (A) the spectrum of A EM n (R) _ The matrix A EM n (R) is called (positive) stable if AEa (A) implies Re (A) > 0_ We shall denote the multiplicative group of diagonal matrices with positive diagonal ' entries in M n (R) by D,,_ DEFINITION: AEMn(R) is called D-stable ifDA is stable for all DEDn. Several sufficient and some necessary conditions for D-stability are known; however, no general characterization is yet known _ In this note we present conditions on the D-stable matrices in Mn(R) when n=2, 3, and 4_ Only one of the known necessary conditions will be of interest to us here_ DEFINITION: AEMn(R) belongs to the class Po [2] if and only iffor each k=l, ___ , n all k by k principal minors of A are nonnegative_ If also, at least one principal minor of each order is positive, then AEP6_ The best necessary condition for D-stability seems to be THEOREM 0: [4,5] If AEMn(R) is D-stable, then AEP't. The converse of theorem 0 is, in general, far from valid_ However, for n=2 we have THEOREM 1: AEMz(R) is D-stable if and only if AEPt. PROOF: The necessity follows from theorem O. Suppose A EPt n M 2 (R) and that D is an arbitrary element of D z. Then DA has positive trace and positive determinant. Since DA EM 2 (R), this means that DA is positive stable and that A is D-stable which completes the proof. For our remaining work we shall employ the stability theorem of Routh and Hurwitz [3]. For AEMn(R) denote the sum of the (~) principal minors of order k by EdA). Define the Routh-Hurwitz matrix fl by

Second, third, and fourth order D-stability