Explanation Given: The given matrix A is, A = [ 3 − 1 − 2 1 6 1 1 0 6 ] Approach: Solve the vector differential equation x ' = A x in the case where A is a defective matrix by constructing cycles of generalized eigenvectors and forming corresponding solutions x k ( t ) as, Let, A be an n × n matrix. Suppose the eigenvalue λ has multiplicity m and that the dimension of the corresponding eigenspace is r ≤ m . Then there exist m linearly independent solutions to x ' = A x that can be constructed in r cycles (one for each i with 1 ≤ i ≤ r ) of the form. x 0 ( i ) ( t ) = e λ t v 0 ( i ) , x 1 ( i ) ( t ) = e λ t ( v 1 ( i ) + t v 0 ( i ) ) x 2 ( i ) ( t ) = e λ t ( v 2 ( i ) + t v 1 ( i ) + 1 2 ! t 2 v 0 ( i ) ) ⋮ x k i ( i ) ( t ) = e λ t ( v k i ( i ) ) + t v k i − 1 ( i ) + ⋯ + 1 k ! t k i v 0 ( i ) Where each k i ≥ 0 , and k 1 + k 2 + ⋯ + k r = m − r . Calculation: Consider the given matrix A , A = [ 3 − 1 − 2 1 6 1 1 0 6 ] The characteristic equation of A is given as det ( A − λ I ) = 0 , det ( A − λ I ) = 0 | [ 3 − 1 − 2 1 6 1 1 0 6 ] − [ λ 0 0 0 λ 0 0 0 λ ] | = 0 | [ 3 − λ − 1 − 2 1 6 − λ 1 1 0 6 − λ ] | = 0 Solve the above equation for the eigenvalues ( λ ) of A , | 3 − λ − 1 − 2 1 6 − λ 1 1 0 6 − λ | = 0 The eigenvalues of A are 5 , 5 , 5 . The eigenvalue of the given matrix is 5 with a multiplicity of 3 . The eigenvector of the this eigenvalue will be of the form v = r [ − 1 , 0 , 1 ] T . The solution to x ' = A x is of the form x 0 ( t ) = e 5 t v 0 . The two linearly independent solutions will be of the form x 1 ( t ) = e 5 t ( v 1 + t v 0 ) and x 2 ( t ) = e 5 t ( v 2 + t v 1 + 1 2 ! t 2 v 0 ) where, ( A − 5 I ) 3 v 3 = 0 or ( A − 5 I ) 2 v 2 ≠ 0 And, v 1 = ( A − 5 I ) v 2 v 0 = ( A − 5 I ) 2 v 2 . We get A − 5 I = [ − 2 − 1 − 2 1 1 1 1 0 1 ] , ( A − 5 I ) 2 = [ 1 1 1 0 0 0 − 1 − 1 − 1 ] and ( A − 5 I ) 3 = [ 0 0 0 0 0 0 0 0 0 ] , choose any vector v 2 such that it satisfies ( A − 5 I ) 3 v 3 = 0 and ( A − 5 I ) 2 v 2 ≠ 0 . Let, v 2 = [ 1 0 0 ]

Differential Equations and Linear Algebra (4th Edition)

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ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 9.11, Problem 18AP

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The general solution to the system x'=Ax for the given matrix A.

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Differential Equations and Linear Algebra (4th Edition)

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