I am trying to find an orthogonal matrix $P$ of the quadric $3x^2+3z^2-4xy-8xz+4yz=1$.
I have found that the eigenvalues are $-1$ (with algebraic multiplicity 2) and $8$ (with algebraic multiplicity 1) with corresponding eigenvectors {<1,0,1>,<-0.5,-2,0.5>} and {<2,-1,-2>} respectively. I know, once normalised, that these eigenvectors form the columns of $P$, but must the $det(P)=1$? If so, my matrix $P$ is incorrect as I can't manipulate the eigenvectors to adhere to this property. This is a question from a past exam paper and help on this problem would be greatly appreciated
We just need to normalize the vectors by $$u_i=\frac{v_i}{|v_i|}$$
in order to obtain $P$ such that $P^TP=I$.