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Non-symmetric but positive definite

Source: 2018 South Korea USCM P7

August 14, 2020
linear algebrapositive definitecollege contestsmatrixvectoreigenvalue

Problem Statement

Suppose a 3×33\times 3 matrix AA satisfies vtAv>0\mathbf{v}^t A \mathbf{v} > 0 for any vector v∈R3−{0}\mathbf{v} \in\mathbb{R}^3 -\{0\}. (Note that AA may not be a symmetric matrix.) (1) Prove that det⁡(A)>0\det(A)>0. (2) Consider diagonal matrix D=diag(−1,1,1)D=\text{diag}(-1,1,1). Prove that there's exactly one negative real among eigenvalues of ADAD.
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