(a) Let K M be positive definite n n matrices and 1 n be their generalized eigenvalues, as in Exercise 8 4 9 Prove that that the largest generalized eigenvalue can be characterized by the maximum principle 1 max xTKx xTMx 11 (b) Prove the alternative maximum principle (c) How would you characterize the smallest generalized eigenvalue (d) An intermediate generalized eigenvalue x' K x A, max x 0 x' M x

Question: (a) Let K . M be positive definite n n matrices and 1 ... n be their generalized eigenvalues, as in Exercise

(a) Let K . M be positive definite n Ã— n matrices and Î»1 ‰¥ ... ‰¥ Î»n be their generalized eigenvalues, as in Exercise 8.4.9. Prove that that the largest generalized eigenvalue can be characterized by the maximum principle
Î»1 = max {xTKx | xTMx = 11}.
(b) Prove the alternative maximum principle

x' K x A, = max x + 0 x' M x

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