# Of section the solution of the homogeneous system is

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Unformatted text preview: ————————————————————— CHAPTER 7. —— . The characteristic equation of the coefficient matrix is single root of multiplicity three, . Setting , we have , with a . The system of algebraic equations reduce to a single equation An eigenvector vector is given by Since the last equation has two free variables, a second linearly independent eigenvector associated with is Therefore two solutions are obtained as x . It follows directly that x satisfy A and x Hence the coefficient vectors must . Rearranging the terms , we have A A I Given an eigenvector , it follows that A A I I . . . Note that a linear combination of two eigenvectors, associated with the same eigenvalue, is also an eigenvector. Consider the equation A I The augmented matrix is Using elementary row operations, we obtain It is evident that a solution exists provided . Let . . The components of the generalized eigenvector must satisfy ____________________________________________________________...
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