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