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Finding eigenvectors of a simple 2x2 matrix

1. The problem statement, all variables and given/known data

The matrix is:
|1 2|
|3 4|


3. The attempt at a solution

I've worked out the eigenvalues to be [tex]\stackrel{\underline{5\pm\sqrt{33}}}{2}[/tex]

But when I plug the first eigenvalue back in I get:

|1 - [tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]........................2 |
|3........................4 - [tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex] |

Multiplied by (x,y) = (0,0)

[Sorry I couldn't fit that on the same line as the matrix.]


Which in turn gives two equations:

x*(1-[tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]) + 2y = 0

and


3x + y*(4-[tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]) = 0


Which I can't work out how to solve for the eigenvector.

Is the only solution x=y=0?
 

Dick

Science Advisor
Homework Helper
26,255
618
You solve them like you solve any two linear equations. Remember if (x,y) is an eigenvector then a*(x,y) is also an eigenvector. So you are going to have an undetermined parameter which you may as well may make x. Put say, x=1 and solve for y. You should get the same value of y from both equations, since one equation is really a multiple of the other.
 
Thanks for the reply!

Thats exactly what I've been trying to do but X and/or Y always end up canceling and I just get some useless number.
 

Dick

Science Advisor
Homework Helper
26,255
618
Thanks for the reply!

Thats exactly what I've been trying to do but X and/or Y always end up canceling and I just get some useless number.
Are you trying to eliminate a variable from the two equations and solve for the other? That won't work. If you eliminate x then the resulting equation is 0*y=0. Is that your problem? Like I said before, that's happening because the two equations are really the same. They have an infinite number of solutions. Put x equal to any constant (like x=1) and then solve for y.
 
AH!

Thank you!
 

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