Further Maths Question about eigenvectors. Help needed

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sleepysmurf
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The vector e is an eigenvector of the square matrix G. Show that

i) e is an eigenvector of G+kI where k is a scalar and I is an identity matrix

ii) e is an eigenvector of G^2 ->(G "squared")
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cayley-hamilton
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Originally posted by sleepysmurf
The vector e is an eigenvector of the square matrix G. Show that

i) e is an eigenvector of G+kI where k is a scalar and I is an identity matrix

ii) e is an eigenvector of G^2 ->(G "squared")

(i) (pretend each letter has a squiggly line under it apart from the constants lambda, k and mu, nu)

By definition G.e=(lambda).e (1)

therefore (G+kI).e=G.e+k.I.e (multiplying out)
= (lambda).e + k.e (using 1)
=(k+lambda).e

ie. new scalar constant is (k+lambda) say 'mu'

(ii) using 1

G.G.e=G.(lambda).e (post mutliplying by G)
=(lambda).G.e (position of contant doesn't matter
=(lamda).(lambda).e (subs G.e+(lambda.e from 1)
= ((lamda)^2).e

Therefore:

(G^2).e=((lamda)^2).e

ie. new scalar constant is (k+lambda) say 'nu'
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cayley-hamilton
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sorry last sentence should read:

"ie. new scalar constant is (lamda^2) say 'nu'"
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