A Criterion for Stability of Matrices

Ž . Let A be an n = n matrix and let s A be its spectrum. The stability Ž . Ž .4 modulus of A is s A s max Re l: l g s A , and A is said to be stable Ž . if s A 0. The stability of a matrix is related to the Routh]Hurwitz problem on the number of zeros of a polynomial that have negative real parts. Much research has been devoted to the latter. The first solution w x dates back to Sturm 21, p. 304 . Using Sturm’s method, Routh developed a simple algorithm to solve the problem. Hurwitz independently discovered necessary and sufficient conditions for all of the zeros to have negative real parts, which are known today as the Routh]Hurwitz conditions. A good and concise account of the Routh]Hurwitz problem can be found in w x 5 . According to the Routh]Hurwitz conditions, a 2 = 2 real matrix A is Ž . Ž . stable if and only if tr A 0 and det A ) 0; a 3 = 3 real matrix A is Ž . Ž . Ž . Ž . stable if and only if tr A 0, det A 0, and tr A ? a det A , where 2 a is the sum of all 2 = 2 principal minors of A. 2