Unformatted text preview: responding to this value of ! because
_dim N!A " ! 3 ! 1 ! 2_________.
!2 3
Those corresponding eigenvectors are: v 1 !
" 0 , v2 !
" 1 3 (any two independent 0 vectors in the null space will do )
The characteristic polynomial must have a factor of __! 2 __________ because _! ! 0 has
multiplicity at least two due to two independent eigenvectors for ! ! 0________ .
The characteristic polynomial of A is p!! " ! __! 2 !8 ! ! "__________________ (calculate
below).
3!! 2 !1 6 4!! !2 !3 !2 1!! det ! 8! 2 ! ! 3 The remaining eigenvalue is _! ! 8______ and the corresponding eigenvector is
!1
__v 3 !
"
!2 _. (Show your work below)
1
Add the second row to the first below to aid in calculations:
!5 2 !1 6 1 !4 !2 # !3 !2 !7 !2 !3 6 !4 !2 !3 !2 !7 1 !2 !3
# 0 1 2 0 0 0 1 !2
# !3 0 16 8 0 !8 !16 101
# 012
000 2 3. (6 pts)
!1 101
If I tell you that the eigenvectors of A ! are 0
, 0 101
1
0
can you easily check? What eigenvalues correspond to these eigenvectors? , how 1 010 0 , 1 1 Take each proposed eigenvector and multiply by A :
101 !1 !1 010 0 101 1 101 0 010 1 101 0 0 0 101 1 2 1 010 0 101 1 0
! 0 !0 0 1 0
! ! 1 0
2 so ! 1 ! 0 0 0
!1 !2 1 0 so ! 2 ! 1 so ! 3 ! 2 1 What eigenvalues correspond to these eigenvectors? Given that A is symmetric, it should
have a complete set of orthonormal eigenvectors? What vectors (explicitly) are those...
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This document was uploaded on 02/19/2014 for the course MATH 441 at WVU.
 Spring '08
 STAFF
 Math, Linear Algebra, Algebra

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