I'm having difficulties in finding the diagonal forms of some quadratic forms. I am sure it is not supposed to be that difficult but I guess I am lacking some creativity after overdoze of coffee and lack of sleep... Here is one quadratic form that I really need some help with.
$q = x_{1}x_{2} + x_{2}x_{3} + ... +x_{n-1}x_{n}$
I see that if q were $q = x_{1}x_{2} + x_{3}x_{4} + ... + x_{2n-1}x_{2n}$, then it would be an orthogonal sum of hyperbolic planes and hence it would be a hyperbolic space. But this one has a repetition of variables... so I am lost.
Would you please help me from this madness????