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