Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}\dfrac{ 2 }{ 5 }&\dfrac{ 17 }{ 5 }\\-\dfrac{ 1 }{ 5 }&\dfrac{ 4 }{ 5 }\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ -{{4i-3}\over{5}} $ | $ \begin{bmatrix} 1 \cr -{{4~i-1}\over{17}} \end{bmatrix} $ |
| $ {{4i+3}\over{5}} $ | $ \begin{bmatrix} 1 \cr {{4~i+1}\over{17}} \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = \lambda^2-\frac{ 6 }{ 5 }\lambda+1 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ \lambda^2-\frac{ 6 }{ 5 }\lambda+1 = 0. $
The eigenvalues are
$$ \lambda_1 = -{{4i-3}\over{5}} ~ , ~ \lambda_2 = {{4i+3}\over{5}} $$( 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 = -{{4i-3}\over{5}} $ we have $ A x = -{{4i-3}\over{5}} I $.
For $ \lambda_ 1 = {{4i+3}\over{5}} $ we have $ A x = {{4i+3}\over{5}} I $.
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