I'd like suggestions of books that address the general equation of the quadrics ($Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$), that is, a book that teach to rotate a quadric and also teach to find what quadric is when it's given rotated .
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Suggest Treil Linear Algebra Done Wrong.
Next, suggest you add another variable, say $w,$ to arrive at $$ A x^2 + B y^2 + C z^2 + Dxy + E zx + F yz + G xw + H yw + I zw + J w^2. $$ The Hessian matrix, $M,$ of second partial derivatives for this is $$ M = \left( \begin{array}{cccc} 2A & D & E & G \\ D & 2B & F & H \\ E & F & 2C & I \\ G & H & I & 2 J \end{array} \right) $$ Finally, either for the original polynomial or for this, there is no guarantee of a closed form (radicals) for the eigenvalues. However, these are not necessary. If all the (10?) coefficients are integers, it is guaranteed that there is an invertible matrix $P$ with all coefficients integer or rational, such that $$ P^T M P = N, $$ where $N$ is diagonal. The diagonal elements of $N$ tell you all you need to know about the type of quadric, even though there is distortion involved. That is, the diagonal entries of $N$ are not the eigenvalues of anything. Finally, there is a simple algorithm for finding such a matrix $P,$ both in the Treil book and at links in reference for linear algebra books that teach reverse Hermite method for symmetric matrices
Oh, if you use all four variables, at the end you set $w=1.$
Here is a 4 by 4 with bad eigenvalues: Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it? I give an explicit $P$ in my answer there.
