**((users:au:index_e Author))** --- **((:algebra2:optimiz:distance:vspom2 Русская версия))** ---- !!§!! Satellite page for ☞ ((:matricese:optimize:distancee#distance_between_quadrics Distance evaluation between geometric objects)) ---- We consider here the manifolds in $ \mathbb R^{n} $. !!T!! **Theorem.** //Let the matrix// $ {\mathbf A}_1 $// be positive definite. The quadrics// $$ X^{\top}{\mathbf A}_1X=1 \quad and \quad X^{\top}{\mathbf A}_2X=1 $$ //do not intersect iff the matrix// $ {\mathbf A}_1 - {\mathbf A}_2 $ //is sign-definite.// **Proof.** Let there exist a point $ X=X_0 \in \mathbb{R}^n $ such that $ X_0^{\top} {\mathbf A}_1X_0=1 $ and $ X_0^{\top} {\mathbf A}_2X_0=1 $. Thus $ X_0^{\top} ( {\mathbf A}_1 -{\mathbf A}_2) X_0=0 $ for $ X_0 \ne \mathbb{O} $. Therefore, $ {\mathbf A}_1 - {\mathbf A}_2 $ is not sign-definite. Conversely, if $ {\mathbf A}_1 - {\mathbf A}_2 $ is not sign-definite then there exists $ X_0 \ne \mathbb{O} $ such that $ X_0^{\top} ( {\mathbf A}_1 -{\mathbf A}_2) X_0=0 $. Alternatively, $ X_0^{\top}{\mathbf A}_1X_0=X_0^{\top}{\mathbf A}_2X_0 $. Multiply the latter by a scalar $ t^2 $ with $ t \in \mathbb{R} $: $$ t^2X_0^{\top}{\mathbf A}_1X_0=t^2X_0^{\top}{\mathbf A}_2X_0 .$$ Set $$ t=1/\sqrt{X_0^{\top}{\mathbf A}_1X_0} $$ (the radicand is positive due to the positive definiteness of $ {\mathbf A}_1 $). The point $ X=tX_0 $ is an intersection point for both manifolds since $$X^{\top}{\mathbf A}_1X=t^2X_0^{\top}{\mathbf A}_1X_0=1 \mbox{ and } X^{\top}{\mathbf A}_2X=t^2X_0^{\top}{\mathbf A}_2X_0=t^2X_0^{\top}{\mathbf A}_1X_0=1\, .$$ !!=>!! If $ n_{} $ is even and $ \det (A_2-A_1)<0 $ then ellipsoid $ X^{\top} A_{1} X =1 $ intersects the quadric $ X^{\top} A_{2} X =1 $. !!=>!! If the ellipsoid $ X^{\top} A_{1} X =1 $ touches the quadric $ X^{\top} A_{2} X =1 $ at least in one point then $ \det (A_2-A_1) = 0 $. == References == I failed to find the original sourse for the result. It is contained as a problem in the problem book [1]. **Proskuryakov, I.V.** //Problems in Linear Algebra.// Mir. Moscow. 1978