Eigenvectors - EIGENVECTORS Method for solving systems of first order differential equations using eigenvalues and eigenvectors 9 p288 x1 = 2 x1 5x 2 x

# Eigenvectors - EIGENVECTORS Method for solving systems of...

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EIGENVECTORS Method for solving systems of first order differential equations using eigenvalues and eigenvectors 9. p288 x x x 1 1 2 2 5 ' = - x x x 2 1 2 4 2 ' = - x x 1 2 0 2 0 3 ( ) , ( ) = = The problem can be rewritten x x ' = - - 2 5 4 2 Definition: An eigenvalue of matrix A is a number l such that A I - = l 0 To find the eigenvalues we rewrite the matrix 2 5 4 2 - - - - l l and find its determinant ( )( ) ( )( ) 2 2 5 4 - - - - - l l which we simplify and set equal to zero l 2 16 0 + = Solving for l we get l = ± - 0 0 64 2 yielding complex eigenvalues l = ± 0 4 i Definition: An eigenvector associated with the eigenvalue l is a nonzero vector v such that Av v = l so that ( ) A I v - = l 0 To find the eigenvectors we set up the equation 2 4 5 4 2 4 0 0 - - - - = ( ) ( ) i i a b using the eigenvalue 0 4 + i Since in this case we have complex eigenvalues it will be necessary to find only one eigenvector to solve the problem. Otherwise we would need to repeat these steps using the second eigenvalue to find a second eigenvector.