matrix with cos and sin, eigenvectors,values and formula?

fiksi · **Joined:** Sun, 8 Feb 2009 22:02:17 UTC **Posts:** 21

Hi guys, this question bothers me...

I have a matrix A=

cos(theta) sin(theta)
-sin(theta) cos(theta)

So it is a 2x2 matrix. I am asked to show that e to the power of (i theta)
and e to the power of (-i theta) are eigenvalues. Then to find eigenvectors and simple formula for A to pwr of n.

Now, I guess I should proceed finding determinant of A(with - lambda from left top to bottom right)=0. However, i can't find my way out of this. I get smth like lambda squared + sin squared theta+ cos squared theta - 2lambda cos theta=0.

Any help is appreciated!

Matt · **Joined:** Wed, 1 Oct 2003 04:45:43 UTC **Posts:** 9631

fiksi wrote:

I get smth like lambda squared + sin squared theta+ cos squared theta - 2lambda cos theta=0.

This can be rewritten as $\lambda^2-(2\cos\theta)\lambda+1=0.$ Now find the roots; use the quadratic formula.

fiksi · **Joined:** Sun, 8 Feb 2009 22:02:17 UTC **Posts:** 21

Matt wrote:

fiksi wrote:

I get smth like lambda squared + sin squared theta+ cos squared theta - 2lambda cos theta=0.

This can be rewritten as $\lambda^2-(2\cos\theta)\lambda+1=0.$ Now find the roots; use the quadratic formula.

Thx, yes, might have missed smth obvious here... but as soon as I saw sin and cos, I didn't like the look of it... so I can use quadratic equation, but what bothers me is how to even get to e to the power itheta now. I vaguely remember there was some identity about it, not sure now.

Matt · **Joined:** Wed, 1 Oct 2003 04:45:43 UTC **Posts:** 9631

fiksi wrote:

but what bothers me is how to even get to e to the power itheta now. I vaguely remember there was some identity about it, not sure now.

Euler's formula: $e^{i\theta}=\cos\theta+i\sin\theta$

fiksi · **Joined:** Sun, 8 Feb 2009 22:02:17 UTC **Posts:** 21

Matt wrote:

fiksi wrote:

but what bothers me is how to even get to e to the power itheta now. I vaguely remember there was some identity about it, not sure now.

Euler's formula: $e^{i\theta}=\cos\theta+i\sin\theta$

Yes, thx once again... I used this when calculating eigenvectors again, got M matrix, and then a simple formula for A to pwr of n.

The final solution to A to pwr of n contains just sin and cos (n theta), right? I got smth like that... hope it's correct.

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matrix with cos and sin, eigenvectors,values and formula?

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