Find the signature of quadratic form : $\left(x^2+y^2+z^2\right)-4\left(xy+xz+yz\right)$

Question

If anyone can help I would like to know what is the signature of the following quadratic form :

$$f(x,y,z) = \left(3x^2+4y^2+z^2\right)-9\left(5xy+3xz+yz\right)$$

A few steps would be a lot appreciated thanks in advance for your help.

Answer 1

Hint

$$f(x,y,z)=\begin{pmatrix}x&y&z\end{pmatrix}\begin{pmatrix}1&-2&-2\\-2&1&-2\\-2&-2&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}.$$

Then try to diagonalize the matrix (at least, finding the eigenvalue is enough).