Question
Find eigenvalues of matrix:
$$ A = \left[ \begin{matrix}0&0&-\frac{ 1 }{ 2 }\\0&0&0\\-\frac{ 1 }{ 2 }&0&0\end{matrix} \right] $$Answer
The eigenvalues of matrix A are:
$$ \lambda_1 = 0 ~ \text{ , } ~ \lambda_2 = \dfrac{ 1 }{ 2 } ~ \text{ and } ~ \lambda_3 = -\dfrac{ 1 }{ 2 } $$Explanation
Step 1 : The characteristic polynomial for matrix A is:
$$ p(\lambda) = -\lambda^3+\frac{ 1 }{ 4 }\lambda $$
Step 2 : Eigenvalues are roots of the characteristic polynomial, so the equation
$$ -\lambda^3+\frac{ 1 }{ 4 }\lambda = 0 $$yields the above eigenvalues.
( click here to view an explanation on how to solve this equation.)
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