(1) Let A be 3x3 real symmetric matrix. The eigenvalues of $A$ are $\lambda_1 = -6, \lambda_2 = 1, \lambda_3=4$
$q(x_1,x_2,x_3) = -x_1^2 + x_2^2 -x^2_3 + 10x_1x_3 = 1$. $A$ is the matrix of $q$.
I want to find the closest point to the origin. I write $q$ as $-6c_1^2 + 1 +c_2^2 + 4c_3^2 = 1$, where $c_i$ is $x_i$ written according to the orthonormal basis eigenbasis $B$ of $A$.
I've been told that this occurs when $c_1=c_2 =0$. But this is far from obvious to me.
(2) Lastly, is it possible to get multiple answers to this question because you could have said that the eigenvalues are 4,-6,1, in which case you would have the equation $4c_1^2 - 6c_2^2 + c_3^2 = 1$. Wouldn't this give you a different answer?