A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C ₁, C ₂, C ₃ be three symmetric matrices in ℜ ^n×n , if max{x ^T C ₁ x,x ^T C ₂ x,x ^T C ₃ x}≥0 for all x∈ℜ ⁿ , it is proved that there exist t _i ≥0 (i=1,2,3) such that ∑ _i=1 ³ t _i =1 and ∑ _i=1 ³ t _i C _i has at most one negative eigenvalue. Received February 18, 1997 / Revised version received October 1, 1997? Published online June 11, 1999