Minimize a quadratic equation with diagonalization

Question

I have a problem to solve and I have no idea how to do. I need to find the maximum and minimum of $G=2x²+2y²-15z²+8xy-12yz-12xz$ with $x²+y²+z²=1$ I'm supposed to use the diagonalization but I don't know with which matrix started... Thanks for your help :)

Answer 1

hint: Lagrange Multiplier will work for this question ! Let $H(x,y,z) = x^2+y^2+z^2-1$, and solve $\nabla G = \alpha \nabla H$. You solve for $x,y,z, \alpha$.

Answer 2

Hint: You can write G = $x' \cdot Q \cdot x$ where $x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ and $Q = \begin{bmatrix} 2 & 4 & -6 \\ 4 & 2 & -6 \\ -6 & -6 & -15 \end{bmatrix} $