含多个参数的实齐二次型的正定性

Positive Definiteness of Real Homogeneous Quadratic Form with Several Parameters

本文给出了一类含多个参数的实齐二次型Π_r(x,φ_1,…,φ_r),φ_i=ξ_ic_i^Tx,在给定参数范围(?)≤ξ_i≤K_i内均为正定二次型的条件。

Let x∈Rn and i∈R (i = 1, … ,r) and let IIr(x, 1,…,r) be a real homogeneous quadratic form with respect to x, 1,…,r.. Now write downorDefinition: >0 or it is positive definite if every IIr(ξi)∈ is positive definite (the same for M>0).The main results will be:Theorem 1. Let IIr∈D, then >0 or it is positive definite if and only if M is positive definite or M>0