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 Post subject: matrix with cos and sin, eigenvectors,values and formula?Posted: Sun, 8 Feb 2009 22:15:12 UTC
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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!

Last edited by fiksi on Mon, 10 May 2010 15:44:34 UTC, edited 1 time in total.

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 Post subject: Re: matrix with cos and sin, eigenvectors,values and formulaPosted: Sun, 8 Feb 2009 23:14:50 UTC
 Member of the 'S.O.S. Math' Hall of Fame

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 Now find the roots; use the quadratic formula.

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 Post subject: Re: matrix with cos and sin, eigenvectors,values and formulaPosted: Sun, 8 Feb 2009 23:45:49 UTC
 Member

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 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.

Last edited by fiksi on Mon, 10 May 2010 15:44:34 UTC, edited 1 time in total.

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 Post subject: Posted: Sun, 8 Feb 2009 23:49:03 UTC
 Member of the 'S.O.S. Math' Hall of Fame

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:

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 Post subject: Posted: Thu, 12 Feb 2009 17:58:54 UTC
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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:

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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