I have a problem with determining eigenvalues. This is what I've got thus far:

Identify and sketch the graph of the quadratic equation

4x² + 10xy + 4y² = 9

[This gives the matrix form:

$\displaystyle \begin{pmatrix} 4 & 5 \\

5 & 4 \\

\end{pmatrix}$

Now we find the eigenvalues:

Det(A – x

*I*[font=Arial]) = $\displaystyle \begin{pmatrix} (4-x) & 5 \\

5 & (4-x) \\

\end{pmatrix}$

= x² – 8x – 9

= (x – 9)(x + 1)

eigenvalues are $\displaystyle \lambda1 = 9 and + \lambda1 = -1$

From there, it's pretty simple solving:

$\displaystyle \lambda1x'^2 + \lambda2y'^2 = 9$

My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.