# Conic Sections with Matrices

I'm reading "Geometry" by Brannan, Esplen, and Gray. Any conic has an equation of the form $$Ax^{2} + Bxy + Cy^{2} + Fx + Gy + H = 0,$$ where $A,\ B,\ C,\ F,\ G,\ H\ \in {\mathbb R}$ and not all of $A, B, C$ are zero. The matrix form of this equation is $$x^{\sf T}{\cal A}x + J^{\sf T}x + H = 0,$$ where $x = \begin{bmatrix} x \\ y\end{bmatrix}$ is a vector in $\;\Bbb R^2\;$, ${\cal A} = \begin{bmatrix} A & B/2 \\ B/2 & C\end{bmatrix}$, and $J = \begin{bmatrix} F \\ G\end{bmatrix}$. Given the equation $$3x^2 - 10xy + 3y^2 + 14x -2y + 3 = 0,$$ we are asked to find what type of conic this is and its center (if it has one).

You start by diagonalizing the matrix $A$ and constructing an orthogonal matrix, $P$, with the normalized eigenvectors of $A$. Now, the book says that it's important to make sure that columns of $P$ are arranged s.t. ${\rm det}\ (P) = 1$, so $P$ represents a rotation in the plane. My question is, why is it necessary that $P$ represents a rotation of $x$? Why can't ${\rm det}\ (P) = -1$, so $P$ represents a reflection followed by a rotation? I tried seeing where this change would cause a problem, but can't see it. Can anyone please clarify?

The curve is identified as a conic that has been rotated from the standard position. Having the determinant $+1$, the new axes are 'labeled' so that, visibly they are obtained by rotating the old ones by an angle (that can be worked out by looking at $P$).

So it is all about the axes. Nothing much will go wrong if you don't bother checking the determinant of $P$ (often enough students are not even told about this, and they put whatever orientation on the axes).

• I see. Thank you for the explanation. – Adam Nov 29 '13 at 5:27

The short answer is, since conic sections are symmetric, if a reflection followed by rotation exists to get a result, then just a rotation also exists to get the result.

The longer answer is similar to @Any's answer. For whatever application you are using a determinant, the sign almost always reflects some choice of which direction you put positive and negative on your axises. A conic section is a conic section regardless of which direction you put the signs of the axises, so we expect the sign of the determinant to be insignificant.

Remark: consider doing yourself a huge favor and use $$Ax^2 + 2Bxy + Cy^2 + 2Fx + 2Gy + H = 0$$ as your canonical form for a conic section. It saves you lots of trouble from having to carry lots of around unnecessary coefficients/fractions when working out problems.

• Thank you for the explanation and tip :) – Adam Nov 29 '13 at 5:27
