soln4b

# soln4b - MATH 235 Solutions Assignment 4b 1 Let A be an n...

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Unformatted text preview: MATH 235 Solutions Assignment 4b 1. Let A be an n × n matrix. Also, let α, β, γ be three distinct eigenvalues of A having corresponding eigenvectors x , y , z , respectively. Consider the vector v = x + y + z . Can v be an eigenvector of A corresponding to an eigenvalue λ (possibly different from α, β, γ ? Explain. Solution : Given A ⇀ x = α ⇀ x, A ⇀ y = β ⇀ y , A ⇀ z = γ ⇀ z and ⇀ v = ⇀ x + ⇀ y + ⇀ z , then A ⇀ v = A ⇀ x + A ⇀ y + A ⇀ z = α ⇀ x + β ⇀ y γ ⇀ z . (1) Assume ⇀ v is an eigenvector of A corresponding to the eigenvalue λ . Then A ⇀ v = λ ⇀ v . From (1), it follows that α ⇀ x + β ⇀ y + γ ⇀ z = λ parenleftBig ⇀ x + ⇀ y + ⇀ z parenrightBig ⇒ ( α- λ ) ⇀ x +( β- λ ) ⇀ y +( γ- λ ) ⇀ z = ⇀ . (2) Since the eigenvectors ⇀ x, ⇀ y , ⇀ z belong to distinct eigenvalues, the eigenvectors ⇀ x, ⇀ y , ⇀ z are linearly independent. Thus, (2) gives that α- λ = β- λ = γ- λ = 0 ⇒ α = β =...
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## This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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soln4b - MATH 235 Solutions Assignment 4b 1 Let A be an n...

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