MATH152 April 2010
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Question B 02 (a)
is known to have eigenvalues 1 and 4.
(a) Find the eigenvector of A that corresponds to eigenvalue 4.
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Recall the eigenvectors with eigenvalue satisfy
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We want to find the eigenvector of
with eigenvalue . We therefore want to find the vector such that
We therefore have three equations with three unknowns
We can solve the first equation for c to get
and can sub this into the second equation to get
Using c=2a-b and b=a we conclude that c=a. From the last equation we get,
which is satisfied for all a. Therefore, let a=1 without loss of generality. This implies that b=c=1 as well. Therefore the eigenvector for the eigenvalue 4 is
Of course, any multiple of this is also an eigenvector (expect multiplying by 0).