Explanation Consider the given system of equation as A = ( − 3 0 2 1 − 1 0 − 2 − 1 0 ) x . Note that, a single vector equation is x ′ = A x , where A is coefficient matrix . The coefficient matrix of the corresponding systems of differential equations is A = ( − 3 0 2 1 − 1 0 − 2 − 1 0 ) . Characteristic equation of the coefficient matrix A = ( − 3 0 2 1 − 1 0 − 2 − 1 0 ) is det ( A − λ I ) = 0 , where λ eigenvalue and I is 3 × 3 identity matrix . Simplify for the det ( A − λ I ) = 0 using the coefficient matrix as follows. det ( A − λ I ) = 0 det ( ( − 3 0 2 1 − 1 0 − 2 − 1 0 ) − λ ( 1 0 0 0 1 0 0 0 1 ) ) = 0 det ( − 3 − λ 0 2 1 − 1 − λ 0 − 2 − 1 − λ ) = 0 ( − 3 − λ ) | − 1 − λ 0 − 1 − λ | + 2 | 1 − 1 − λ − 2 − 1 | = 0 Further simplified as, ( − 3 − λ ) ( λ + λ 2 ) + 2 ( 1 + 2 λ ) = 0 − 3 λ − 3 λ 2 − λ 2 − λ 3 + 2 + 4 λ = 0 λ 3 + 4 λ 2 − λ − 2 = 0 ( λ + 2 ) ( λ + 1 − i 2 ) ( λ + 1 + i 2 ) = 0 λ = − 2 , λ = − 1 + i 2 , λ = − 1 − i 2 Compute the eigenvector x ( 1 ) = ( t 1 t 2 t 3 ) corresponding to the eigenvalue λ = − 2 as follows. A x − ( − 2 ) x = 0 ( − 3 + 2 0 2 1 − 1 + 2 0 − 2 − 1 2 ) ( t 1 t 2 t 3 ) = ( 0 0 ) ( − 1 0 2 1 1 0 − 2 − 1 2 ) ( t 1 t 2 t 3 ) = ( 0 0 ) Use row reduction to obtain the eigen vectors as follows. A − λ I = ( − 1 0 2 1 1 0 − 2 − 1 2 ) ∼ ( 1 0 − 2 0 1 2 − 2 − 1 2 ) ( − R 1 → R 1 − R 1 + R 2 → R 2 ) ∼ ( 1 0 − 2 0 1 2 0 0 0 ) ( 2 R 1 + R 3 → R 3 R 1 + R 3 → R 3 ) From above matrix, it is observed that the system of equation is { x 1 − 0 x 2 − 2 x 3 = 0 0 x 1 + x 2 + 2 x 3 = 0 0 x 1 + 0 x 2 + 0 x 3 = 0 . So, the value of x 1 2 = x 3 1 and the value of x 2 − 2 = x 3 1 . Hence, the possible eigenvector corresponding to eigenvalue λ = − 2 is x ( 1 ) = ( 2 − 2 1 ) . Compute the eigenvector x ( 2 ) = ( t 1 t 2 t 3 ) corresponding to the eigenvalue λ = − 1 − 2 i as follows. A x − ( − 1 − 2 i ) x = 0 ( − 3 + 1 + i 2 0 2 1 − 1 + 1 + i 2 0 − 2 − 1 1 + i 2 ) ( t 1 t 2 t 3 ) = ( 0 0 ) ( − 2 + i 2 0 2 1 i 2 0 − 2 − 1 1 + i 2 ) ( t 1 t 2 t 3 ) = ( 0 0 ) Use row reduction to obtain the eigen vectors as follows

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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