# help with eigenvalue proof

• May 6th 2009, 07:11 PM
chuckienz
help with eigenvalue proof
Show that if 0 < θ < π then the matrix $\displaystyle B=\left(\begin{array}{cc}cos\theta&-sin\theta\\sin\theta&cos\theta\end{array}\right)$
has no real eigenvalues and consequently no eigenvectors.
• May 6th 2009, 07:47 PM
TheEmptySet
Quote:

Originally Posted by chuckienz
Show that if 0 < θ < π then the matrix $\displaystyle B=\left(\begin{array}{cc}cos\theta&-sin\theta\\sin\theta&cos\theta\end{array}\right)$
has no real eigenvalues and consequently no eigenvectors.

$\displaystyle \left(\begin{array}{cc}cos\theta -\lambda &-sin\theta\\sin\theta&cos\theta -\lambda \end{array}\right)$

So we take the determinate of the above matrix to get

$\displaystyle (\cos(\theta)-\lambda)^2+\sin^2(\theta)=0$

$\displaystyle \cos^2(\theta)-2\lambda \cos(\theta)+\lambda^2+\sin^2(\theta)=0$

$\displaystyle \lambda^2-2\cos(\theta) \lambda+1=0$

This will only have real solutions if the discriminate is posative

So the discriminate of this quadratic is

$\displaystyle (-2\cos(\theta))^2-4(1)(1)=4\cos^{2}(\theta)-4=4(\cos^2(\theta)-1)$

This is negative for $\displaystyle 0< \theta < \pi$

So the equation have no Real solutions
• May 6th 2009, 10:52 PM
Moo
FYI, its eigenvalues are $\displaystyle e^{i\pi}~,~ e^{-i\pi}$
• May 6th 2009, 11:17 PM
Gamma
Quote:

Originally Posted by Moo
FYI, its eigenvalues are $\displaystyle e^{i\pi}~,~ e^{-i\pi}$

$\displaystyle e^{i\pi}=-1=e^{-i\pi} \in \mathbb{R}$
• May 6th 2009, 11:21 PM
Moo
Quote:

Originally Posted by Gamma
$\displaystyle e^{i\pi}=-1=e^{-i\pi} \in \mathbb{R}$

I meant theta :D

$\displaystyle e^{i\theta}~,~ e^{-i\theta}$
• May 6th 2009, 11:26 PM
Gamma
ahhh! Okay, I should have realized that is what you meant. Sorry