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Unformatted text preview: Complex Eigenvalues Background on Complex Eigenvectors/Eigenvalues Recall that given a matrix A , we said that a nonzero vector v was an eigenvector of A with eigenvalue if A v = v where is a scalar. Nothing about this definition required that be a real number, nor that v have real number components. But since we have seen that eigenvalues arise as the roots of the polynomial equation det( A- I ) = 0 and there is no reason to expect that the roots of this equation will not in some cases be complex numbers, as the following example shows. I Example Find the eigenvalues of the matrix A =- 1 1 and for each eigenvalue find a corresponding eigenvector. Solution The eigenvalues are the roots of the equation det( A- I ) = 0 det- - 1 1- = 0 2 + 1 = 0 = i. Thus our eigenvalues are 1 = i and 2 =- i . To find an eigenvector for 1 = i we need to solve ( A- 1 I ) v = 0 - i- 1 1- i v 1 v 2 = - iv 1- v 2 = 0 v 1- iv 2 = 0 The second equation is just...
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