Given any quadric can be represented in matrix form, assume I have one quadric $Q_1$ and another quadric $Q_2$. How can I find the equation of the surface that defines intersection of $Q_1$ and $Q_2$ (assuming it exists) using matrix algebra?
For example, the intersection of two spheres is an ellipse in 3d space with an equation I can write as $f(x,y,z) = c$. Is it possible to find the intersection of $\mathbf{x}^t Q_1 \mathbf{x} = 0$ and $\mathbf{x}^t Q_2 \mathbf{x} = 0$ without needing to expand these equations and solve algebraically? i.e. working just with the matrices $Q_1$ and $Q_2$?