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Question

# Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.$\left( \begin{array}{rrrr}{-2} & {1} & {0} & {0} \\ {1} & {0} & {0} & {1} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$

Solution

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For given matrix

$A=\left(\begin{array}{cccc} -2 & 1 & 0 & 0\\ 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right)$

the characteristic polynomial is:

$p_A(\lambda)=|\lambda\boldsymbol{I_4}-\boldsymbol{A}|=\left|\begin{array}{cccc} \lambda+2 & -1 & 0 & 0\\ -1 & \lambda & 0 & -1\\ 0 & 0 & \lambda & 0\\ 0 & 0 & 0 & \lambda \end{array}\right|=\lambda^2(\lambda(\lambda+2)-1)=\lambda^2(\lambda^2+2\lambda-1).$

Its roots (the eigenvalues of the matrix $A$) are $\color{#c34632}\lambda_1=\lambda_2=0,\;\lambda_3=-1+\sqrt{2}$ and $\color{#c34632}\lambda_4=-1-\sqrt{2}$.

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