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MAT211_Lecture18[1]

# MAT211_Lecture18[1] - Consider the projection P on R2 onto...

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MAT211 Lecture 18 Eigenvalues and eigenvectors Consider the projection P on R 2 onto the x-axis. Find all vectors v such that P(v) is parallel to v. Consider the reflection on R 2 with respect to the x-axis. Find all vectors v such that R(v) is parallel to v. Definition Consider an n x n matrix A. A vector v in Rn is an eigenvector if Av is a multiple of v, that is, if there exists a scalar k such that Av=kv. A scalar k such that Av=kv for some vector v is an eigenvalue . Example Consider an orthogonal projection onto a plane P on R 3 . Find all the eigenvalues and eigenvectors. Consider reflection with respect to a plane P on R 3 . Find all the eigenvalues and eigenvectors. Consider reflection with respect to a line L on R 3 . Find all the eigenvalues and eigenvectors. 5 Example • Consider the matrix of a rotation of angle ! /3 in R 2 . Find all the eigenvalues and eigenvectors. What are the eigenvalues and eigenvectors of any rotation?

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