Question

Eigenvectors:
I have a matrix
(-6 -10)
( 5 9)

I have calculated the correct eigenvalues of 4 and -1
However when I calculate the eigenvectors I am getting my postive/negative the wrong way round, could someone please explain where I am going wrong please?

Thanks!

Answer 1

As a guess... without working it... I would say that is because when going from Eigenvalues to Eighenvectors, order matters.

Answer 2

Eigenvectors are really about lines, where positive and negative aren't as important, the eigenvalues determine if the magnitudes are increasing or decreasing. so (-1,2) is the same as (1,-2) because they describe the same 'line." So the independent EVs are like basis functions (similar to x,y-axis). Projections of vectors onto these vectors occurs correctly so long as you are aware of the direction of your EV. So if you project and your EV is pointing in a certain direction, you know which direction because you defined what "positive" means.

Answer 3

Okay, once I sub in my eigenvalues, I get:
For
\[\lambda = 4 \]\[\left[\begin{matrix}-10 & -10 \\ 5 & 5\end{matrix}\right]\]
\[\lambda = -1\] \[\left[\begin{matrix}-5 & -10 \\ 5 & 10\end{matrix}\right]\]

How do you go about finding the eigenvectors, ie what method, I thought I understood it but I don't think I do now...

Thanks for your help :)

Answer 4

Oh it's okay, I've just redone the question and I know where I went wrong!

Thanks for the help :D

Answer 5

Kk. Have fun! So where did you go wrong?