Question
Find eigenvectors of matrix:
$$ A = \left[ \begin{matrix}75&0&-60\\0&50&0\\-60&0&-15\end{matrix} \right] $$Answer
The eigenvectors and eigenvalues of matrix A are:
| Eigenvalue | Eigenvector |
| $ 105 $ | $ \begin{bmatrix} 1 \cr 0 \cr -{{1}\over{2}} \end{bmatrix} $ |
| $ -45 $ | $ \begin{bmatrix} 1 \cr 0 \cr 2 \end{bmatrix} $ |
| $ 50 $ | $ \begin{bmatrix} 0 \cr 1 \cr 0 \end{bmatrix} $ |
Explanation
Step 1 : Find characteristic polynomial $ p(\lambda) $:
$$ p(\lambda) = -\lambda^3+110\lambda^2+1725\lambda-236250 $$Step 2 : Find the eigenvalues by solving the characteristic equation $ -\lambda^3+110\lambda^2+1725\lambda-236250 = 0. $
The eigenvalues are
$$ \lambda_1 = 105 ~ , ~ \lambda_2 = -45 ~ , ~ \lambda_3 = 50 $$( 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 = 105 $ we have $ A x = 105 I $.
For $ \lambda_ 1 = -45 $ we have $ A x = -45 I $.
For $ \lambda_ 2 = 50 $ we have $ A x = 50 I $.
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