Prove: If xT Ax is a quadratic form whose minimum and maximum values subject to the

Chapter 7, Problem 22

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Prove: If xT Ax is a quadratic form whose minimum and maximum values subject to the constraint x = 1 aremandM,respectively, then for each number c in the interval m c M, there is a unit vector xc such that xT c Axc = c. [Hint: In the case where m < M, let um and uM be unit eigenvectors of A such that uT mAum = m and uT M AuM = M, and let xc = , M c M mum + , c m M muM Show that xT c Axc = c.]