/sci/ - Science & Math

Are there any fundamental theorems left to explore in mathematics?

/sci/ do you have any tips and tricks on determining eigenvectors more easier?

I am solving the linear equation for a damped harmonic oscillator (2nd order, linear). I have reduced it to a 1st order problem and now I'm solving x' - Ax = 0, where A is a DQ system.

I know I can solve it in a "normal" way without all this fancy Linear Algebra stuff, but I think it's fun.

So the matrix A looks like this
http://www.wolframalpha.com/input/?i={{0%2C1}%3B{-w^2%2C-2y}}

I got those eigenvalues, for lamba_1 I arrive at this matrix
http://www.wolframalpha.com/input/?i={{y-sqrt%28y^2-w^2%29%2C1}%3B{-w^2%2C-y-sqrt%28y^2-w^2%29}}

Whose eigenvectors I like to calculate. Obviously wolframalpha has already given me the answer, but I can't always use that.

What are some fast ways or tips and tricks that help "seeing" one what the Eigenvectors of a certain matrix might be?
>gif not related

>>6343032
We both know I'm right, but it's okay shorty, I don't mean to hurt your feelings

just do me a favor and never call it a science again, it rustles my jimmies
*hug*

>mfw I'm studying god tier and shit tier at the same time

(Mathematics and Philosophy)
p.s: Philosophy is NOT shit tier, unless you're attempting to find a job.