(a) Let K . M be positive definite n n matrices and 1 ... ...

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(a) Let K . M be positive definite n × n matrices and λ1 ‰¥ ... ‰¥ λn be their generalized eigenvalues, as in Exercise 8.4.9. Prove that that the largest generalized eigenvalue can be characterized by the maximum principle
λ1 = max {xTKx | xTMx = 11}.
(b) Prove the alternative maximum principle
x' K x A, = max x + 0 x' M x

(c) How would you characterize the smallest generalized eigenvalue?
(d) An intermediate generalized eigenvalue?

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Applied Linear Algebra

ISBN: 978-0131473829

1st edition

Authors: Peter J. Olver, Cheri Shakiban

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