eigen values ch6.pptx - Eigenvalues and Eigenvectors 1-Definitions Definition 1 A nonzero vector x is an eigenvector(or characteristic vector of a

eigen values ch6.pptx - Eigenvalues and Eigenvectors...

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Eigenvalues and Eigenvectors
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1 -Definitions Definition 1: A nonzero vector x is an eigenvector (or characteristic vector ) of a square matrix A if there exists a scalar λ such that Ax = λx . Then λ is an eigenvalue (or characteristic value ) of A . Note : The zero vector can not be an eigenvector even though A0 = λ0 . But λ = 0 can be an eigenvalue. Example: Show x 2 1 isaneigenvector for A 2 4 3 6 Solution : Ax 2 4 3 6 2 1 0 0 But for 0, x 0 2 1 0 0 Thus , xisaneigenvector of A , and 0 isaneigenvalue .
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2- Eigenvalues Let x be an eigenvector of the matrix A. Then there must exist an eigenvalue λ such that Ax = λx or, equivalently, Ax - λx = 0 or (A – λI)x = 0 If we define a new matrix B = A – λI , then Bx = 0 If B has an inverse then x = B -1 0 = 0 . But an eigenvector cannot be zero. Thus, it follows that x will be an eigenvector of A if and only if B does not have an inverse, or equivalently det(B)=0 , or det(A – λI) = 0 This is called the characteristic equation of A . Its roots determine the eigenvalues of A .
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