Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}0&-3&1&2\\-2&1&-1&2\\-2&1&-1&2\\-2&-3&1&4\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ 2 $ | $ \begin{bmatrix} 1 \cr 0 \cr 0 \cr 1 \end{bmatrix} $ |
| $ 2 $ | $ \begin{bmatrix} 1 \cr 0 \cr 0 \cr 1 \end{bmatrix} $ |
| $ 0 $ | $ \begin{bmatrix} 0 \cr 1 \cr 1 \cr 1 \end{bmatrix} $ |
| $ 0 $ | $ \begin{bmatrix} 0 \cr 1 \cr 1 \cr 1 \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = \lambda^4-4\,\lambda^3+4\,\lambda^2 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ \lambda^4-4\,\lambda^3+4\,\lambda^2 = 0. $
The eigenvalues are
$$ \lambda_1 = 2 ~ , ~ \lambda_2 = 2 ~ , ~ \lambda_3 = 0 ~ , ~ \lambda_4 = 0 $$( 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 = 2 $ we have $ A x = 2 I $.
For $ \lambda_ 2 = 0 $ we have $ A x = 0 I $.
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