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#### karush

##### Well-known member

- Jan 31, 2012

- 3,066

1. having two distinct real eigenvalues,

$A=\begin{bmatrix}

2 & -1\\

-1& 2

\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & -1 \\ -1 & 2 - \lambda \end{array} \right|=\lambda^{2} - 4 \lambda + 3 \therefore \lambda_1=3\ \lambda_2=1$

2. a pair of complex eigenvalues.

$\left| \begin{array}{cc}

2 - \lambda & 1 \\

-1 & 2 - \lambda

\end{array} \right|=\lambda^{2} - 4 \lambda + 5\quad \lambda_{1}=2 - i,\lambda_{1}=2 + i$

3. two identical real eigenvalues,

ok hopefully 1 and 2 are ok

on 3 I was just going to do this $(\lambda-2)(\lambda-2)$

and go backwards to a matrix but ?????