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Unformatted text preview: P ( x ) = ax 2 + bx + c . If A is an n × n matrix, deﬁne P ( A ) = aA 2 + bA + cI . Prove that if λ is an eigenvalue of A then P ( λ ) is an eigenvalue of P ( A ). If λ is an eigenvalue of A then there is a nonzero vector v satisfying Av = λv . Thus, P ( A ) v = aA 2 v + bAv + cIv = λaAv + bλv + cv = aλ 2 v + bλv + cv = ( aλ 2 + bλ + c )( v ) = P ( λ ) v . This shows that v is also an eigenvector of P ( A ) with eigenvalue P ( λ ) proving that P ( λ ) is an eigenvalue of P ( A ). 2...
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 Fall '10
 yildiz
 Math, Linear Algebra, λ, Mike Hartglass

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