# Canonical form of a degenerate conic

I have the conic $\Gamma:x^2+y^2-2xy-6=0$, which is degenerate since \begin{equation} \begin{vmatrix}1&-1&0\\-1&1&0\\0&0&-6\end{vmatrix}=0 \end{equation} and particular it represents two parallel lines as the rank of the complete matrix is 2(the second row is equal to the first one with inverted signs).

The exercise ask me the canonical form and gives the result $X^2-3=0$ but I've never did any exercise nor given any thought on how to write the canonical form of a degenerate conic. Can someone explain to me please?

## 1 Answer

You can easily detect the binomial theorem inside the formula, giving $$(x-y)^2-6=0$$ $x-y=(1,-1)\dbinom{x}{y}$, and normalizing the first factor as part of an orthogonal matrix gives the equivalent form $$2X^2-6=0,\qquad \begin{bmatrix}X\\Y\end{bmatrix} =\frac1{\sqrt2}\begin{bmatrix}1&-1\\1&1\end{bmatrix} ·\begin{bmatrix}x\\y\end{bmatrix}$$ Now divide by $2$.

• I think I have understood your explaination but still I want to be sure. One of the way to find the canonical form of a conic is through traslation+rotation. The traslation is written as $X=x-x_{0}\wedge Y=y-y_{0}$. The passage matrix you wrote is $\begin{bmatrix} \cos(\theta)&\sin(\theta)\\ -\sin(\theta)&\cos(\theta) \end{bmatrix}$ with $\theta=\frac{\pi}{4}$. By normalizing by dividing for $|(1,-1)|=\sqrt(2)$ and multiplying it for $(x,y)^T$ you get $(X,Y)^T$. – M.Giacchello Dec 5 '16 at 10:04
• You have to find an orthogonal vector. $[1,1]$ is orthogonal to $[1,-1]$, now normalize again. In the end you want to decompose $B=Q^TDQ$ with $Q$ orthogonal and $D$ diagonal, $B$ is your original symmetric matrix which has an eigen decomposition with simple real eigenvalues and orthogonal eigenvectors by the spectral theorem. – LutzL Dec 5 '16 at 10:08
• My professor has not taught us about orthogonal matrices nor spectral theorem though. So I have to solve this excercise with the rototraslation method by explaining the conceptual steps. So even if there's clearly a link with what you are saying I cannot use your explaination and I can't fully grasp it, I'm sorry. I really appreciate you help though, I'll try to study your answer. – M.Giacchello Dec 5 '16 at 10:17
• You will only get a normal form by rotations in special, constructed cases. So do not be discouraged if this method does not work outside textbook exercises. In general, the eigenvalues, and consequently the entries of the rotation matrix, even if real, are not nice numbers, they are roots of polynomials of degree equal the matrix dimension. – LutzL Dec 5 '16 at 11:31