What am I doing wrong on this linear question

[bluewhistled]

Homework Statement

The matrix is [[5,2],[2,2]], the eigenvalues are 6 and 1, the eigenvectors are [2/(5^.5),1/(5^.5)] and [1/(5^.5),-2/(5^.5)]

The Attempt at a Solution

I'm trying to reproduce their results and I can get the eigenvalues and the first vector but the second one always comes out to be [-.5,1]. when it should be [.5,-1] according to the book. Unless the book is wrong (which I'm hoping someone will help me verify So anyway this is what I end up with [2,1] and [-.5,1] which you normalize to get the above answers but the second vector has the signs reversed. I'll type up my work so you can see if there is anything wrong with it...

(L-5)(L-2)-4=L^2-7L+6=(L-6)(L-1)
Which gives L=1,6

Now for L=1:
v_1=[[-4,-2],[-2,-1]] which simplifies to [[1,-2],[0,0]] and then I take the eigenvector of [2,1]

v_2=[[1,-2],[-2,4]] which simplifies to [[1,.5],[0,0]] and then I take the eigenvector of [-.5,1]

Can anyone see a flaw in my work?

Thanks a lot

 

[Redbelly98]

You and they are both correct. Multiplying an eigenvector by any scalar produces the "same" eigenvector solution. That scalar multiple can be -1.

 

[bluewhistled]

Oh lol, now I feel like a fool for stressing out for an hour at 3AM last night over this. Thanks a lot for clearing that up.