Finding the eigenvectors (and behavior of solution) around the

[Somefantastik]

finding the eigenvectors (and behavior of solution) around the critical points found in this thread: https://www.physicsforums.com/showthread.php?t=258349&referrerid=110346

$$D_{f} = \[\begin{pmatrix}32x & 18y \\ 32x & -32y\end{pmatrix}\]$$

$$D_{f}(1,1) = \[\begin{pmatrix}32 & 18 \\ 32 & -32\end{pmatrix}\]$$

$$= \[\begin{pmatrix}16 & 9 \\ 16 & -16 \end{pmatrix}\]$$

$$det(A-\lambda I) =\[\begin{pmatrix} 16-\lambda & 9 \\ 16 & -16- \lambda \end{pmatrix}\]$$

$$= -256 + \lambda^{2} - 146 \ => \ \lambda = ^{+}_{-}20$$

$$\lambda_{1} = 20:$$

$$(A-\lambda_{1} I)\xi^{(1)} = 0 \ => \ \[\begin{pmatrix} -4 & 9 \\ 16 & -36 \end{pmatrix}\]\xi^{(1)} = 0$$

I can't get LaTeX to cooperate with me, that's supposed to say [-4 9; 16 -36]ξ⁽¹⁾ = 0

$$=> \ \xi^{(1)} = \left[^{9}_{4} \right]$$

Having trouble finding $$\xi^{(2)}$$ when $$\lambda_{2} = -20$$.

Keeps coming out to be [0 0]^T.

Any suggestions?

 

[Somefantastik]

Looks like you can't reduce the matrix before you do det(A - tI). Figured it out; thanks for looking.