Problem 1288 (difficulty: 5/10)

Let \(\displaystyle A\) and \(\displaystyle B\) be \(\displaystyle n\times n\) real symmetric matrices where \(\displaystyle \det A\ne0\).

(a) Prove that if \(\displaystyle x\to x^TBx\) has a local extremum at \(\displaystyle x_0\in\R^n\) given the constraint \(\displaystyle x^TAx=1\), then \(\displaystyle x_0\) is an eigenvector of \(\displaystyle A^{-1}B\).

(b) What is the meaning of the eigenvalue corresponding to the eigenvector \(\displaystyle x_0\)?

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