The inertia of diagonal multiples of 3×3 real matrices

AbstractTo each triple (α, β, γ) of non-negative integers satisfying α + β + γ = 3 there corresponds the class of 3×3 real matrices M such that the inertia In (MD) = (α, β, γ) for every 3×3 positive definite diagonal matrix D. Each such class is characterized by giving algebraic conditions which the principal minors of its members satisfy. These characterizations are obtained as corollaries of a general theorem on the roots of real homogeneous polynomials of order 3 and degree 3, and they make it possible to characterize for 3×3 matrices (1) those M such that In(MD) = In (D) for all diagonal D and (2) those M such that MD is stable if and only if D is stable. The latter is the n = 3 case of the original definition of D-stability due to Arrow and McManus [1] and Enthoven and Arrow [3]