Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}2&0&-2\\0&0&-2\\-2&-2&1\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ -2 $ | $ \begin{bmatrix} 1 \cr 2 \cr 2 \end{bmatrix} $ |
| $ 4 $ | $ \begin{bmatrix} 1 \cr {{1}\over{2}} \cr -1 \end{bmatrix} $ |
| $ 1 $ | $ \begin{bmatrix} 1 \cr -1 \cr {{1 }\over{2}} \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = -\lambda^3+3\lambda^2+6\lambda-8 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ -\lambda^3+3\lambda^2+6\lambda-8 = 0. $
The eigenvalues are
$$ \lambda_1 = -2 ~ , ~ \lambda_2 = 4 ~ , ~ \lambda_3 = 1 $$( 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 = -2 $ we have $ A x = -2 I $.
For $ \lambda_ 1 = 4 $ we have $ A x = 4 I $.
For $ \lambda_ 2 = 1 $ we have $ A x = 1 I $.
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