Suppose we have the equation \[ \langle a - Ax, B(a - Ax) \rangle \leq t \]
where $a,x \in \mathbb{R}^n$ and $A,B \in \mathbb{R}^{n\times n}$ are invertible. Is there a way I could solve the above to get something like \[ x \in E \] where $E$ is the ellipsoid defined by the quadratic equation?
Let $u=A^{-1}a$, then: \begin{align}⟨a−Ax,B(a−Ax)⟩&=(-1)^2⟨A(x-A^{-1}a),BA(x-A^{-1}a)⟩ \\ &=(x-u)^TA^TBA(x-u) \\ &=(x-u)^T\frac 12(A^TBA+(A^TBA)^T)(x-u) \\ &=(x-u)^T\frac 12(A^T(B+B^T)A)(x-u) \\ &=(x-u)^TUDU^T(x-u) \\ &\le t \end{align} This is a quadratic form like an ellipsoid, translated by $u$, transformed by the orthogonal matrix $U$, and identified by the diagonal matrix $D$.