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7/25/2019 Math Zc234 l8
1/22
BITSPilaniHyderabad Campus
BITSPilaniHyderabad Campus
Dipak Kumar Satpathi
Department of Mathematics
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 " #$%( '
Linear Algebra & Optimization MATH ZC234 )
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E-
Linear Algebra & Optimization MATH ZC234 *
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Linear Algebra & Optimization MATH ZC234 +
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Linear Algebra & Optimization MATH ZC234 ,
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Linear Algebra & Optimization MATH ZC234 -
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Linear Algebra & Optimization MATH ZC234 .'
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Linear Algebra & Optimization MATH ZC234 ..
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Linear Algebra & Optimization MATH ZC234 .2
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Linear Algebra & Optimization MATH ZC234 .3
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E-.
Linear Algebra & Optimization MATH ZC234 .)
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Linear Algebra & Optimization MATH ZC234 .*
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Note that eigenvalues are ) ) /
Linear Algebra & Optimization MATH ZC234 .+
ro+ reduce the au!mented matri, for
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Linear Algebra & Optimization MATH ZC234 .,
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E-ample
0et
'ind the eigenvalues of A. 'ind theeigenspace of A.
Linear Algebra & Optimization MATH ZC234 2'
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Linear Algebra & Optimization MATH ZC234 2.
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Linear l!ebra " #ptimi$ation M%H &C'() ''
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