Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}5&2&1\\0&10&15\\0&0&32\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ 5 $ | $ \begin{bmatrix} 1 \cr 0 \cr 0 \end{bmatrix} $ |
| $ 10 $ | $ \begin{bmatrix} 1 \cr {{5}\over{2}} \cr 0 \end{bmatrix} $ |
| $ 32 $ | $ \begin{bmatrix} 1 \cr {{405 }\over{52}} \cr {{297}\over{26}} \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = -\lambda^3+47\lambda^2-530\lambda+1600 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ -\lambda^3+47\lambda^2-530\lambda+1600 = 0. $
The eigenvalues are
$$ \lambda_1 = 5 ~ , ~ \lambda_2 = 10 ~ , ~ \lambda_3 = 32 $$( 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 = 5 $ we have $ A x = 5 I $.
For $ \lambda_ 1 = 10 $ we have $ A x = 10 I $.
For $ \lambda_ 2 = 32 $ we have $ A x = 32 I $.
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