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Unformatted text preview: Lecture 14 Eigenvalues and Eigenvectors Suppose that A is a square ( n n ) matrix. We say that a nonzero vector v is an eigenvector ( ev ) and a number is its eigenvalue ( ew ) if A v = v . (14.1) Geometrically this means that A v is in the same direction as v , since multiplying a vector by a number changes its length, but not its direction. Matlab has a built-in routine for finding eigenvalues and eigenvectors: > A = pascal(4) > [v e] = eig(A) The results are a matrix v that contains eigenvectors as columns and a diagonal matrix e that contains eigenvalues on the diagonal. We can check this by: > v1 = v(:,1) > A*v1 > e(1,1)*v1 Finding Eigenvalues for 2 2 and 3 3 If A is 2 2 or 3 3 then we can find its eigenvalues and eigenvectors by hand. Notice that Equation (14.1) can be rewritten as: A v- v = . (14.2) It would be nice to factor out the v from the right-hand side of this equation, but we cant because A is a matrix and is a number. However, since I v = v , we can do the following: A v- v = A v- I v = ( A- I ) v = (14.3) If v is nonzero, then by Theorem 3 in Lecture 10 the matrix ( A- I ) must be singular. By the same theorem, we must have: det( A- I ) = 0 . (14.4) This is called the characteristic equation ....
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