(a) Show that if A is a real symmetric matrix and n is its

Partial Differential Equations: An Introduction | 2nd Edition | ISBN: 9780470054567 | Authors: Walter A. Strauss

ISBN: 9780470054567 222

Solution for problem 9 Chapter 11

Partial Differential Equations: An Introduction | 2nd Edition

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Problem 9

(a) Show that if A is a real symmetric matrix and n is its largest eigenvalue, then n =max c=0 Acc |c|2 . (b) IfBisanotherrealsymmetricmatrix,Bispositivedenite,and n is thelargestrootofthepolynomialequationdet(AB)=0,showthat n =max c=0 Acc Bcc . [Hint for (b): Use the fact that B has a unique square root that is positive denite symmetric.]

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/t =(Y),,/ ->Y'/r E.- e . I -i fv,, +rTl t71{l,lf b tr'- qb+ h9- lr Wc'' ,J Q-- 'sJ+ 'Ch--qft-_''lj (sr/ ) =O1' Lq t0lJ \ -= \ Q=X Q = hf \' o 1oz- Q= \/a+ x)XC- Q= X-(-,\ 'L/t - ' xzl +,\g _-e).,rf \{=-\Ua-z\4 G),J=(.:l J :r]^rn\ y.naQa;u1 1.

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