Consider the function $$ f(x, y, z) := x^2 + y^2 + z^2 - \sqrt{2} x y + \sqrt{2} y z $$ Find an orthogonal $3 \times 3$ matrix $S$ and $\alpha, \beta, \gamma \in \Bbb R$ with $$f(S(x, y, z)^t) = \alpha x^2 + \beta y^2 + \gamma z^2$$ Note: First write $f$ in the form $f(x) = (Ax) - x$ with asymmetrical $3 \times 3$ matrix $A$.
I tried and came to following representation for the function as matrix:
\begin{equation} A= \begin{pmatrix} 1 & -\frac{1} {\sqrt2} & 0\\ -\frac{1} {\sqrt2} & 1 & \frac{1} {\sqrt2}\\ 0 & \frac{1} {\sqrt2} & 1\\ \end{pmatrix} \end{equation}
\begin{matrix} (A⋅x)^t⋅x = x^2+y^2-\sqrt{2}⋅x⋅y+z^2+\sqrt{2}⋅y⋅z \end{matrix}
This looked good to me. Now, I wanted to calculate the eigenvectors for this :
$$ u_{1} = \left(\begin{matrix} -1 \\ -\sqrt{2} \\ 1 \end{matrix}\right), u_2=\left(\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}\right), u_3=\left(\begin{matrix} -1 \\ \sqrt{2} \\ 1 \end{matrix}\right) $$ However, when I use these vectors as my matrix S, it is not symetric and $$S^TS \neq E$$ So, what should I do instead. According to ChatGPT the values α, β, γ are the eigenvalues (0, 1, 2). But as it is right now, I don't think this is correct. I am not really going anywhere since then, so I wanted to ask what to do now/instead. $$ \ $$ Thank you!
Your approach is esentially correct: $S$ is the change of basis matrix from the basis $\{u_1,u_2,u_3\}$ to the standard basis. However, note that $S$ is orthogonal iff the basis $\{u_1,u_2,u_3\}$ is orthonormal. Hence, since your $u_i$’s are already orthogonal, the last thing that you need to do is normalize the basis of eigenvectors $\{u_1,u_2,u_3\}$, i.e., taking $u_i/\lVert u_i\rVert$ instead of $u_i$.