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na_lec_3 - 7 Polynomial roots finding 7.1 Observer form...

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1 7. Polynomial roots finding 7.1. Observer form Consider the monic polynomial ( 29 n n n n x x c x c x c c x P + + + + + = - - 1 1 2 2 1 0 L . Its roots are the eigenvalues of the observer matrix - - - - - = - 1 3 2 1 0 1 0 1 0 1 0 1 0 n c c c c c M O O Z . Proof. The eigenvalues of Z are the roots of the characteristic polynomial ( 29 1 3 2 1 0 1 1 1 1 - - - - - - - - - - - = - = n n n c c c c c Q l l l l l l l M O O I Z Evaluating the determinant using the last column gives ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 l l l l l l l l l l l l l l l l l l l l n n n n n n n n n n n n n n n P c c c c c c c c Q 1 1 1 1 1 1 1 1 1 1 ... 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 3 1 1 2 3 2 2 4 1 2 0 1 2 2 1 1 0 - = - + - + + - + - + - = - - - - - - + + - - - - + - - - - + - - - - = - - - + + - + + L O O O O O O O which completes the proof of the theorem.
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