EigenvectorsDate: 05/13/97 at 22:50:57 From: Nader Iskander Subject: Eigenvectors I have a question on determining any kind of pattern related to eigenvectors. First of all, I do not fully understand what eigenvectors are. All I know is that they can be put in matrix form, and when mutiplied by a point, give a scalar multiple of the original point. If there is more to them, can you please tell me? Thank you very much. Date: 05/14/97 at 16:44:53 From: Doctor Anthony Subject: Re: Eigenvectors The eigenvectors of a matrix are vectors that do not vary in direction when transformed by the matrix. A point on an eigenvector can move up or down the vector when transformed by the matrix, but it will remain on that vector. The scale factor by which such a point moves is given by the associated eigenvalue. If v is an eigenvector and k the associated eigenvalue, then for matrix M, we have: Mv = kv A 3 x 3 matrix will, in general, have 3 linearly independent eigenvectors, each with its associated eigenvalue. If we use these three vectors as the base vectors of the 3D space (that is the position of any point is given in terms of u, v, w the eigenvectors), then on transforming by matrix M, the new position is simply k1*u, k2*v, k3*w where k1, k2, k3 are the 3 eigenvalues, associated respectively with u, v, and w. The eigenvalues and then the eigenvectors are found by solving: |M-k*I| = 0 For a 3 x 3 matrix there will be three roots k1, k2, k3. Using these values in turn in the equation Mv1 = k1*v1, we obtain the three eigenvectors. There are many uses of eigenvectors, a particularly important one is in finding powers of matrices. For example, M^200 is not to be lightly undertaken, but by using the diagonal form of the matrix (in which eigenvalues and eigenvectors play a key role) we can write down the result in a few minutes. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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