# How to simplify the general equation of a conic section by linear algebra?

When encountering the general equation of a conic section $$a_{11}x^2 + a_{12}xy + a_{22}y^2 + b_1x + b_2y + c = 0$$

I can write it in matrix form as a quadratic form of the vector $$(x,y,1)^T$$. But what then? What should be done to reach the form of the standard equation of a conic section, i.e. of an ellipse/hyperbola/parabola with center translated and conic rotated?

You need to compute the eigenvalue decomposition of the quadratic form, which diagonalizes the matrix. So suppose you have $$[x,y,1] A [x,y,1]^T = 0$$ Since $A$ can always be made symmetric, then it should have an eigenvalue decomposition $A=Q\Lambda Q^T$ where $Q$ is orthogonal and $\Lambda$ is diagonal. Then, if you use the rotated coordinates $z=Q[x,y,1]^T$, you get $$z^T \Lambda z = 0$$ which you can multiply out to get the standard form in terms of the components of $z$.