Question
Find eigenvalues of matrix:
$$ A = \left[ \begin{matrix}2&-1&0\\-1&2&-1\\0&-1&2\end{matrix} \right] $$Answer
The eigenvalues of matrix A are:
$$ \lambda_1 = 2 ~ \text{ , } ~ \lambda_2 = 2-\sqrt{ 2 } ~ \text{ and } ~ \lambda_3 = 2+\sqrt{ 2 } $$Explanation
Step 1 : The characteristic polynomial for matrix A is:
$$ p(\lambda) = -\lambda^3+6\lambda^2-10\lambda+4 $$
Step 2 : Eigenvalues are roots of the characteristic polynomial, so the equation
$$ -\lambda^3+6\lambda^2-10\lambda+4 = 0 $$yields the above eigenvalues.
( click here to view an explanation on how to solve this equation.)
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