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lect1123Eigenvalues - MATH 17100 28446 Eigenvalues and...

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MATH 17100 28446 11/23/2009 Eigenvalues and eigenvectors of a Matrix Given a square matrix A , and a vector x , the product Ax is also a vector, which, in general, has a magnitude and direction different from x . In other words, the square matrix A transforms vectors to vectors which, in general, have different orientations. We ask: Given the nn matrix A , are there certain preferred or characteristic directions along which vectors get transformed without change in orientation? In other words, given A , find v such that Av v Vectors oriented along these directions will be transformed into vectors oriented along the same direction with at most a change in magnitude and sign, these vectors, v , are called eigenvectors of A, and the scalar is called the eigenvalue associated with the eigenvector v . Finding the characteristic directions with the associated scalar multiples for a given square
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