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BITSPilaniHyderabad Campus

BITSPilaniHyderabad Campus

Dipak Kumar Satpathi

Department of Mathematics

dipak@hyderabad.bits-pilani.ac.in

MATH ZC234

LINEAR ALGEBRA &OPTIMIZATION

Linear l!ebra " #ptimi$ation M%H&C'()

*

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Eigenvalue & Eigenvector

Linear Algebra & Optimization MATH ZC234 2

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Linear Algebra & Optimization MATH ZC234 3

A matrix eigenvalue problem considers the vector equation

(1) Ax=x.

HereAis a given square matrix,an unknown scalar, andxanunknown vector. In a matrix eigenvalue problem, the task is to

determines andxs that satisfy (1).Sincex=0is always a solution for any and thus not interesting, weonly admit solutions withx0.

The solutions to (1) are given the following names: Thes thatsatisfy (1) are calledeigenvalues of Aand the corresponding

nonzeroxs that also satisfy (1) are calledeigenvectors of A.

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Linear Algebra & Optimization MATH ZC234 4

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characteristic equation of A is

et!A " #$% '

Its roots determine the eigenvalues of A.

Characteristic polynomial of A is et!A " #$%

NOTE: EVE! "#$AE %ATI "ATIE"'IE" IT"C(ACTEI"TIC E#

Once eigenvalues are found) to *nd eigenvectors+e have to solve the homogenous e,uation

!A " #$%( '

Linear Algebra & Optimization MATH ZC234 )

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E-

Linear Algebra & Optimization MATH ZC234 *

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Linear Algebra & Optimization MATH ZC234 +

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Linear Algebra & Optimization MATH ZC234 ,

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Linear Algebra & Optimization MATH ZC234 -

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Linear Algebra & Optimization MATH ZC234 .'

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Linear Algebra & Optimization MATH ZC234 ..

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Linear Algebra & Optimization MATH ZC234 .2

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Linear Algebra & Optimization MATH ZC234 .3

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Linear Algebra & Optimization MATH ZC234 .4

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E-.

Linear Algebra & Optimization MATH ZC234 .)

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Linear Algebra & Optimization MATH ZC234 .*

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Note that eigenvalues are ) ) /

Linear Algebra & Optimization MATH ZC234 .+

ro+ reduce the au!mented matri, for

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Linear Algebra & Optimization MATH ZC234 .,

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Linear Algebra & Optimization MATH ZC234 .-

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E-ample

0et

'ind the eigenvalues of A. 'ind theeigenspace of A.

Linear Algebra & Optimization MATH ZC234 2'

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Linear Algebra & Optimization MATH ZC234 2.

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Linear l!ebra " #ptimi$ation M%H &C'() ''

QUESTIONS?