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bilinear form

Source: miklos schweitzer 2005 q11

August 28, 2021
real analysisHomogeneous functionpositive definitelinear algebra

Problem Statement

Let E:Rn\{0}→R+E: R^n \backslash \{0\} \to R^+ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and p∈Rn\{0}p \in R^n \backslash \{0\} , E(λp)=λ2E(p)E (\lambda p) = \lambda^2 E (p)). Prove that if the second derivative of E′′(p):Rn×Rn→RE''(p): R^n \times R^n \to R is a non-degenerate bilinear form at any point p∈Rn\{0}p \in R^n \backslash \{0\}, then E′′(p)E''(p) (p∈Rn\{0}p \in R^n \backslash \{0\}) is positive definite.
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