Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}0&1&0\\0&0&1\\5&-7&3\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ 1-2i $ | $ \begin{bmatrix} 1 \cr 1-2~i \cr -4~i-3 \end{bmatrix} $ |
| $ 2i+1 $ | $ \begin{bmatrix} 1 \cr 2~i+1 \cr 4~i-3 \end{bmatrix} $ |
| $ 1 $ | $ \begin{bmatrix} 1 \cr 1 \cr 1 \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = -\lambda^3+3\lambda^2-7\lambda+5 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ -\lambda^3+3\lambda^2-7\lambda+5 = 0. $
The eigenvalues are
$$ \lambda_1 = 1-2i ~ , ~ \lambda_2 = 2i+1 ~ , ~ \lambda_3 = 1 $$( click here to view an explanation on how to solve this equation.)
Step 3 : To find the associated eigenvectors ( $ x $ ), we have to solve the equation $ A x = \lambda I $.
For $ \lambda_ 0 = 1-2i $ we have $ A x = 1-2i I $.
For $ \lambda_ 1 = 2i+1 $ we have $ A x = 2i+1 I $.
For $ \lambda_ 2 = 1 $ we have $ A x = 1 I $.
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