Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}10&-9&0&0\\4&-2&0&0\\0&0&-2&-7\\0&0&1&2\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ 4 $ | $ \begin{bmatrix} 1 \cr {{2}\over{3}} \cr 0 \cr 0 \end{bmatrix} $ |
| $ 4 $ | $ \begin{bmatrix} 1 \cr {{2}\over{3}} \cr 0 \cr 0 \end{bmatrix} $ |
| $ -\sqrt{3}i $ | $ \begin{bmatrix} 0 \cr 0 \cr 1 \cr {{\sqrt{3}~i-2}\over{7}} \end{bmatrix} $ |
| $ \sqrt{3}i $ | $ \begin{bmatrix} 0 \cr 0 \cr 1 \cr -{{\sqrt{3}~i+2}\over{7}} \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = \lambda^4-8\,\lambda^3+19\,\lambda^2-24\,\lambda+48 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ \lambda^4-8\,\lambda^3+19\,\lambda^2-24\,\lambda+48 = 0. $
The eigenvalues are
$$ \lambda_1 = 4 ~ , ~ \lambda_2 = 4 ~ , ~ \lambda_3 = -\sqrt{3}i ~ , ~ \lambda_4 = \sqrt{3}i $$( 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 = 4 $ we have $ A x = 4 I $.
For $ \lambda_ 2 = -\sqrt{3}i $ we have $ A x = -\sqrt{3}i I $.
For $ \lambda_ 3 = \sqrt{3}i $ we have $ A x = \sqrt{3}i I $.
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