How can we solve the following equation: $$ x^TAx=k $$ if $k$ = a constant and A is a positive-definite and symmetric matrix, $x$ is a vector. I can clearly see that in 3d we have an infinite set of solutions but I cannot formulate it in a mathematical way. Thanks
2026-03-25 20:34:46.1774470886
Solve $x^TAx=k$ with $k=$cte?
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After diagonalization of the matrix (which is a rotation of the basis), the equation becomes
$$\lambda u^2+\mu v^2+\nu w^2=1,$$
which is the equation of an ellipsoid, as $\lambda,\mu,\nu>0$. (If you make $w$ constant, i.e. cut with a plane, you should recognize the equation of an ellipse.)