- #1

mathmari

Gold Member

MHB

- 5,049

- 7

We consider the surface $S$ of the space $\mathbb{R}^3$ that is defined by the equation $2(x^2+y^2+z^2-xy-xz-yz)+3\sqrt{2}(x-z)=1$.

I want to find (using symmetric matrices) an appropriate orthonormal system of coordinates $(x_1, y_1, z_1)$ for which the above equation has the form $ax_1^2+by_1^2+cz_1^2=d$, for some $a,b,c,d\in \mathbb{R}$.

I have done the following:

The eqquation of the surface $S$ can be written in the form $$\vec{x}^T\begin{pmatrix}2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2\end{pmatrix}\vec{x}+\begin{pmatrix}3\sqrt{2} & 0 & -3\sqrt{2}\end{pmatrix}\vec{x}=1$$ Since that matrix is symmetric it is diagonalizable with orthonormal basis.

Then we have to write the matrix $\begin{pmatrix}2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2\end{pmatrix}$ is the form $PDP^{-1}$, or not?

What do we do next?

(Wondering)