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