Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}-1&2&5\\2&2&2\\5&2&-1\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ -6 $ | $ \begin{bmatrix} 1 \cr 0 \cr -1 \end{bmatrix} $ |
| $ 6 $ | $ \begin{bmatrix} 1 \cr 1 \cr 1 \end{bmatrix} $ |
| $ 0 $ | $ \begin{bmatrix} 1 \cr -2 \cr 1 \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = -\lambda^3+36\lambda $$Step 2 : Find the eigenvalues by solving the characteristic equation $ -\lambda^3+36\lambda = 0. $
The eigenvalues are
$$ \lambda_1 = -6 ~ , ~ \lambda_2 = 6 ~ , ~ \lambda_3 = 0 $$( 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 = -6 $ we have $ A x = -6 I $.
For $ \lambda_ 1 = 6 $ we have $ A x = 6 I $.
For $ \lambda_ 2 = 0 $ we have $ A x = 0 I $.
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