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lect8_8_Eigenvector

lect8_8_Eigenvector - The Eigenvalue Problems 8.8 1...

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The Eigenvalue Problems - 8.8 1. Definition of Eigenvalues and Eigenvectors: Let A be an n ! n matrix. A scalar ! is said to be an eigenvalue of A if the linear system A v " ! v has a nonzero solution vector v . The solution vector v is said to be an eigenvector of A corresponding to the eigenvalue ! . The pair ! , v is called an eigenpair of A . Note that the left side of the equation is a matrix vector multiplication and the right side is a scalar vector multiplication. Example Let A " 2 ! 1 ! 1 2 , u " 1 2 , v 1 " 1 1 , v 2 " 1 ! 1 . Determine if each of u , v 1 , v 2 is an eigenvector of A . If so, what is its eigenvalue? Sketch the graphs of u , v 1 , v 2 and A u , Av 1 , A v 2 . A u " 0 3 " " 1 2 , so u is not an eigenvector of A . A v 1 " 1 1 " v 1 " ! 1 " v 1 , so v 1 is an eigenvector of A corresponding to the eigenvalue ! " 1. A v 2 " 3 ! 3 " 3 1 ! 1 " 3 v 2 , so v 1 is an eigenvector of A corresponding to the eigenvalue ! " 3 -3 -2 -1 0 1 2 3 -3 -2 -1 1 2 3 - u , ... Au -3 -2 -1 0 1 2 3 -3 -2 -1 1 2 3 - v 1 , ... Av 1 -3 -2 -1 0 1 2 3 -3 -2 -1 1 2 3 - v 2 , ... Av 2 How can we find all pairs of eigenvalues and eigenvectors of a given A ? Observe that an eigenpair ! , v satisfies the equation: Av " ! v , that implies that Av ! ! v " ! A ! ! I " v " 0. Since v " 0, the homogeneous system ! A ! ! I " v " 0 has a nonzero solution if and only if the matrix A ! ! I is singular or equivalently det !
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