[tex]\mathbf A=\begin{bmatrix}-10&30\\-6&17\end{bmatrix}[/tex]
Compute the eigenvalues of [tex]\mathbf A[/tex]:
[tex]\begin{vmatrix}-10-\lambda&30\\-6&17-\lambda\end{vmatrix}=(-10-\lambda)(17-\lambda)+180=(\lambda-5)(\lambda-2)=0[/tex]
[tex]\implies\lambda=5,\lambda=2[/tex]
Find the corresponding eigenvectors [tex]\eta[/tex]:
[tex]\lambda_1=2\implies\begin{bmatrix}-12&30\\-6&15\end{bmatrix}\eta_1=\mathbf0[/tex]
[tex]\implies\eta_1=\begin{bmatrix}5\\2\end{bmatrix}[/tex]
[tex]\lambda_2=5\implies\begin{bmatrix}-15&30\\-6&12\end{bmatrix}\eta_2=\mathbf0[/tex]
[tex]\implies\eta_2=\begin{bmatrix}2\\1\end{bmatrix}[/tex]
Now,
[tex]\mathbf A=\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}\mathrm{diag}(\lambda_1,\lambda_2)\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}^{-1}[/tex]
[tex]\mathbf A=\begin{bmatrix}5&2\\2&1\end{bmatrix}\begin{bmatrix}2&0\\0&5\end{bmatrix}\begin{bmatrix}5&2\\2&1\end{bmatrix}^{-1}[/tex]
