MATH 17100 2844611/23/2009 Eigenvalues and eigenvectors of a Matrix Given a squarematrix A, and a vector x, the product Axis also a vector, which, in general, has a magnitude and direction different from x. In other words, the square matrix Atransforms vectors to vectors which, in general, have different orientations. We ask: Given the nnmatrix A, are there certain preferred or characteristic directionsalong which vectors get transformed without change in orientation? In other words, given A, find vsuch that AvvVectors 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 eigenvectorsof A, and the scalar is called theeigenvalueassociated with the eigenvector v. Finding the characteristic directions with the associated scalar multiples for a given square
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