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How to find the eigenvector and determine type, special case.

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Sheng Zang:
I got a question like this,
As I solved, I find the critical point is (0,0) and A= (0                 1)
                                                                          (-6x^2+8     0)
after plug(0,0) into, it becomes (0, 1)
                                              (0, 0)
so, λ1=0=λ2, how to solve eigenvector and determine the type of critical point?

Thanks for help.

Victor Ivrii:
your post is incomprehensible

How to find eigenvector? See Linear algebra course and Sections 7.1, 7.2 of our textbook

Sheng Zang:
I mean forλ1=0, matrix is A=(0 1)  *(x1) =(0)
                                           (0 0)   (y1)    (0)
the eigenvector is (1,0) which is horizontal.

and since λ is repeated, i.e λ1=λ2=0,
we solve (0 1)  *(x2) =(1)
              (0 0)   (y2)    (0)
so, another eigenvector is (0,1), which is vertical.

Am I correct?

Victor Ivrii:
Except $(0,1)$ is not an eigenvector but a generalized e.v.

Sheng Zang:
Also what is the type of critical point?  proper or improper node? And how to draw pic?( I don't think this is as usual as we draw (im)proper node picture)


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