Exercise 766Calculus › Differential equations › Linear systemsL2Added 2 Nov 2026

A system with three distinct eigenvalues

#eigenvalues#eigenvectors#general solution

Statement

Solve $X' = AX$ where
$A = \begin{pmatrix} 0 & 1 & 1 \\ -1 & 2 & 1 \\ 1 & 0 & 1 \end{pmatrix}.$

Hint

The characteristic polynomial is $X(X-1)(X-2)$ . Find the eigenvectors.

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Eigenvalues. $0, 1, 2$ .

Eigenvectors.
- For $\lambda=0$ : $u_0 = (1,1,-1)$ .
- For $\lambda=1$ : $u_1 = (0,-1,1)$ .
- For $\lambda=2$ : $u_2 = (1,1,1)$ .

General solution.
$X(t) = \lambda u_0 + \mu e^t u_1 + \gamma e^{2t} u_2.$