Determine unit normal (eigenvalue, eigenvector)

[TimmvK]

Homework Statement

For a material the stress is defined by the means of the stress matrix O

O = (6 1 -2
1 2 2
-2 2 5) Expressed in MPA

It can be derived that the principe stress are: O₁= 4-sqrt(13), O₂= 5 and O₃=4+sqrt(13)

I know you can derive the principal stresses when you have determined the eigenvalues.

Determine the unit normal N₂ (eigenvector) defining the plane on which the principal stress O₂= 5 [mPA] is acting.

Homework Equations

O.N_i (inproduct)=O_i*N_i for i=1,2,3

I know the answer is N₂= 1/3*sqrt(3)*(e_x+e_y+e_z) But how do I get to this answer? I have no idea. Would be great if someone could help me.

The Attempt at a Solution

 

[vela]

Let A be a matrix and x its eigenvector with eigenvalue λ. Then you have Ax = λx. Rearranging a bit, you get (A-λI)x = 0.

So calculate the matrix A-λI for your system using the appropriate eigenvalue λ and write down the system of equations it corresponds to. The solution to those equations will give you an eigenvector.