eigenvector DE solution workshopGiven the eigenvalues and eigenvectors of the coefficient matrix of a linear homogeneous constant coefficient first order system of vector variable differential equations, 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 transforms via the linear change of variable 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 to 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
where 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 is the diagonalized coefficient matrix,we together go through the steps of writing down the decoupled equations for the new scalar variables, solving them with exponential functions, determining their initial values, and recombining these to get the solutions for the original scalar variables.We will deliver the solution in matrix form, vector form and scalar form:
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
matrix form vector form scalar form
The vector form expresses the solution as a sum of modes, one for each eigenvalue, while evaluating the linear combination or the matrix product leads to the values for the individual variables.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=3x3 matrix case with real distinct eigenvalues;
inits 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Re-order eigenvalues in order from smallest to largest. This part has to be done by hand each time the order changes.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 we fix the result in the order we want.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ivp 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matrix case with a complex conjugate pair of eigenvalues;
inits 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eigenvalues to put real one first, then plus complex 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fix the order we want, real first then complex conjugate 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