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:
Now we find the eigenvalues:
Det(A – xI) =
= x² – 8x – 9
= (x – 9)(x + 1)
eigenvalues are
From there, it's pretty simple solving:
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.
The key point is tracing back the transformation that you have made by diagonalizing the symmetric matrix .
Since are eigenvalues, we find their corresponding eigenvectors. If we do so, we can choose where the indices correspond. Now form the matrix using , i.e., . Let , so . Any solution satisfies the equation . Note that , so . So solutions correspond to solutions of , i.e., the equation , via the transformation .
Hope this helps.
I got it from the quadratic equation:
where a,b,c,d are
I should explain more:
once the equation
has been reached, I need to state what graphic function it is and plot the graph. So it would appear that knowing what and are important as I get either
giving
or
giving
which lead to vastly different graphs. As I said, there's nothing mentioned in my lecture notes about this, and because I'm doing this paper by correspondence I can't go speak to the lecturer.