Homework Help: Finding eigenlines & eigenvalues

1. Feb 3, 2012

Roodles01

In my example I have matrix A = (1 2)
. . . . . . . . . . . . . . . . . . . . . . (3 2)

Finding the eigenvalue through the method I understand & can get the result

i.e.
k = 4 & -1

I suspect my algebra is the shaky link, here, but to find the eigenline I find a bit more of a challenge.

OK I start by substituting the eigenvalue into the eigenvector equation;
Ax = kx
giving
(1 2) (x) = 4 (x)
(3 2) (y)......(y)

which gives rise to the following simultaneous equations
x + 2y = 4x
3x + 2y = 4y

Now the bit I don't get . . .
How do these both reduce to 3x - 2y = 0

I'm sure it's simple & I can't see the wood for the trees, but this is stupidly defeating me.. . . . Grrr!

Please could someone point me in the right direction.

Last edited: Feb 3, 2012
2. Feb 3, 2012

Roodles01

Damn, it's so easy to suddenly see the solution.
OK I'm done

x + 2y = 4x
3x+ 2y =4y

goes to
-3x + 2y = 0
3x - 2y = 0

both reduce to
3x - 2y = 0

Thank you.

3. Feb 5, 2012

Roodles01

All the other examples I have had led to the simultaneous equations reducing to a single equation, from which I can find the eigenline equation for each eigenvalue (2 in this case).

The simultaneous equation I have now is;
3/4 x - 3/4 y = 0
-1/4 x - 1/4 y = 0

I am assuming I have the negative value wrong, but I may not, I suppose, so how can I get back to a single equation so I can find the eigenline equation for each eigenvalue.

4. Feb 5, 2012

HallsofIvy

Please tell us what the matrix in this problem is so that we can tell whether or not 2 really is an eigenvalue.

5. Feb 6, 2012

Roodles01

(11/4 -3/4)
(-1/4 9/4)

This is the original matrix. Thank you.

6. Feb 6, 2012

Deveno

based on the matrix you provided, the second equation should be:

-(1/4)x + (1/4)y = 0.