Quadratic form, determine the surface

[vincentvance]

Hi,

I'm trying to solve this problem and I'm stuck.

What I want to do is determine the kind of surface from this equation:

x²-2y²-3z²-4xy-2xz-6yz = 11

Matrix representation:

1 -2 -1
-2 -2 -3
-1 -3 -3

I want to find the eigenvalues so I write the characteristic equation like this:

1-\lambda -2 -1
-2 -2-\lambda -3
-1 -3 -3-\lambda

(and the goal, of course, is to have this equal to zero to find the eigenvalues - lambda)

Then I want to reduce this to a 2x2 matrix and that's where I get stuck. I've done about 10 other problems similar to this one today and never did I get stuck at this point before, it makes me feel very confused.

Can anyone help me?
Thank you
Vincent

 

[Evgeny.Makarov]

Hi, and welcome to the forum.

1-\lambda -2 -1
-2 -2-\lambda -3
-1 -3 -3-\lambda

(and the goal, of course, is to have this equal to zero to find the eigenvalues - lambda)

Then I want to reduce this to a 2x2 matrix and that's where I get stuck.

I would swap the last two lines (this changes the sign of the determinant) and eliminate the leading $-2$ by subtracting double of line $\begin{pmatrix}-1&-3&-3-\lambda\end{pmatrix}$ from line $\begin{pmatrix}-2&-2-\lambda&-3\end{pmatrix}$.
\[
\begin{pmatrix}
1-\lambda&-2&-1\\
-2&-2-\lambda&-3\\
-1&-3&-3-\lambda
\end{pmatrix}
\mapsto
\begin{pmatrix}
1-\lambda&-2&-1\\
-1&-3&-3-\lambda\\
0&4-\lambda&3+2\lambda
\end{pmatrix}
\]
Then I would expand the determinant along the first column.