- #1
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Construct 3 different $2\times2$ matrices
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 ?
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 ?