Vaki

I'm kind of confused with this excercise:

In 2d space and in respect to an orthonormal system Oxy we are given the curve with equation:

$\displaystyle x^2-10\sqrt{3}xy+11y^2+64=0$

Find an orthonormal system for which the equation takes the simplest form. What is that form?

I'm confused as to what "simplest form" means.. Is it the canonical form? And how do I find the new orthonormal system?

Micrm@ss

Yes, the canonical form.

mathman

The equation looks like an ellipse. Rotate it so there is no xy term.

skipjack

With axes rotated by pi/6, one gets $x^2 - 4y^2 = 16$, the equation of a hyperbola.

(Or one can swap the axes to get $y^2 - 4x^2 = 16$.)

Vaki

I rotated the axis by pi/6 but I found indeed a hyperbola but with this equation instead:

$\displaystyle \frac{3x^2}{32} -\frac{y^2}{4}=1$

Here is what I did:

$\displaystyle x=x'\cos(\frac{\pi}{6})-y'*\sin(\frac{\pi}{6})=\frac{\sqrt3}{2}x'-\frac{1}{2}y'$
$\displaystyle y=x'\sin(\frac{\pi}{6})+y'\cos(\frac{\pi}{6})= \frac{1}{2}x'+\frac{\sqrt3}{2}y'$

I then replace the $\displaystyle x,y$ into the equation which gave me

$\displaystyle \left ( \frac{\sqrt{3}}{2}x'-\frac{1}{2}y' \right )^2-10\sqrt3\left (\frac{\sqrt3}{2}x'-\frac{1}{2}y' \right )\left (\frac{1}{2}x'+\frac{\sqrt{3}}{2}y' \right )+11\left (\frac{1}{2}x'+\frac{\sqrt{3}}{2}y' \right )^2$

And after some calculation, I ended with the final equation:

$\displaystyle \frac{3x'^2}{32} -\frac{y'^2}{4}=1$


Did I miss something?

Country Boy

Since this is in the Linear Algebra sub-forum, write the equation as $\displaystyle \begin{bmatrix} x & y \end{bmatrix}\begin{bmatrix}1 & -5\sqrt{3} \\ -5\sqrt{3} & 11 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}+ 64= 0$.
I have divided the "$\displaystyle -10\sqrt{3}$" terms into the two anti-diagonal terms so that it is a symmetric matrix and has two real eigenvalues.
That matrix has characteristic equation is $\displaystyle \left|\begin{array}{cc}1- \lambda & -5\sqrt{3} \\ -5\sqrt{3} & 11- \lambda\end{array}\right|= \lambda^2- 12\lambda- 64= (\lambda- 16)(\lambda- 4)= 0$ so the eigenvalues are -4 and 16.

To find the eigenvectors corresponding to eigenvalue -4, find x and y such that $\displaystyle \begin{bmatrix}1 & -5\sqrt{3} \\ -5\sqrt{3} & 11 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x- 5\sqrt{3}y \\ -5\sqrt{3}x+ 11y\end{bmatrix}= -4\begin{bmatrix}x \\ y \end{bmatrix}$. The gives the two equations $\displaystyle x- 5\sqrt{3}y= -4x$ and $\displaystyle -5\sqrt{3}x+ 11y= -4y$. Since -4 is an eigenvalue, those two equations are equivalent and reduce to just one equation. For any given y we must have $\displaystyle 5x= 5\sqrt{3}y$ or $\displaystyle x= \sqrt{3}y$. So one eigenvector corresponding to eigenvalue -4 is $\displaystyle \begin{bmatrix}\sqrt{3} \\ 1 \end{bmatrix}$. That has length 2 so a unit eigenvector is $\displaystyle \begin{bmatrix}\frac{\sqrt{3}}{2} \\ \frac{1}{2}\end{bmatrix}$.

Similarly, an eigenvector, $\displaystyle \begin{bmatrix}x \\ y \end{bmatrix}$, corresponding to eigenvalue 16 must satisfy $\displaystyle x- 5\sqrt{3}y= 16x$. For any y we must have $\displaystyle 15x= -5\sqrt{3}y$ or $\displaystyle x= -\frac{\sqrt{3}}{3}a$. So one eigenvector corresponding to eigenvalue 16 is $\displaystyle \begin{bmatrix}-\frac{\sqrt{3}}{3} \\ 1\end{bmatrix}$. That has length $\displaystyle \frac{2}{\sqrt{3}}$ so a unit eigenvector is $\displaystyle \begin{bmatrix}-\frac{1}{2} \\ \frac{\sqrt{3}}{2}\end{bmatrix}$.

Now construct the matrix, P, having those eigenvectors as columns: $\displaystyle P= \begin{bmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}$. Its inverse matrix is $\displaystyle P^{-1}= \begin{bmatrix}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}$, the transpose of P.
P
All of that gives $\displaystyle PAP^{-1}= \begin{bmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}1 & -5\sqrt{3} \\ -5\sqrt{3} & 11 \end{bmatrix}\begin{bmatrix}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}= \begin{bmatrix}-4 & 0 \\ 0 & 16\end{bmatrix}$.

Writing "D" for that diagonal matrix, we can write the original equation, $\displaystyle X^TAX+ 65= 0$ as $\displaystyle X^TP^{-1}PAP^{-1}PX+ 64= (PX)^TD(PX)+ 64= 0$.

With $\displaystyle X= \begin{bmatrix}x \\ y \end{bmatrix}$, $\displaystyle PX= \begin{bmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}\frac{x\sqrt{3}}{2}- \frac{y}{2} \\ \frac{x}{2}+ \frac{y\sqrt{3}}{2}\end{bmatrix}$ so that
$\displaystyle (PX)^TD(PX)+ 64= 0$ becomes $\displaystyle \begin{bmatrix}\frac{x\sqrt{3}}{2}- \frac{y}{2} & \frac{x}{2}+ \frac{y\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}-4 & 0 \\ 0 & 11\end{bmatrix}\begin{bmatrix}\frac{x\sqrt{3}}{2}- \frac{y}{2} \\ \frac{x}{2}+ \frac{y\sqrt{3}}{2}\end{bmatrix}+ 64= 0$.

Finally, if we take $\displaystyle x'= \frac{x\sqrt{3}}{2}- \frac{y}{2}$ and $\displaystyle y'= \frac{x}{2}+ \frac{y\sqrt{3}}{2}$ that will be written as
$\displaystyle -4x'^2+ 16y'^2+ 64= 0$ which, dividing both sides by 64 gives $\displaystyle \frac{y'^2}{4}- \frac{x'^2}{16}= 1$, the equation of a hyperbola.

skipjack

You introduced a sign error in your final step.