Solving for eigenvectors

[Tonig00]

I would like if there is a fast method to find an eigenvector solving a matrix equation directly (knowing a given eigenvalue).

I work in CAS mode:

If we have a matrix a=[[0,1,0,1,0],[0,0,0,0,0],[-1,1,0,2,0],[1,3,0,0,0],[-1,0,1,2,-1]]
We can check with the function "eigenvalue(a)" that -1 is an eigenvalue.

Then if we put the following:
(a*[[x1],[x2],[x3],[x4],[x5]]) = (-1*[[x1],[x2],[x3],[x4],[x5]])

Is there any function that give us a vector like: [[1],[0],[3],[-1],[1]] which fulfils the equation?

The result that give the calculator is: [[(x2+x4) = (-x1)],[0 = (-x2)],[(-x1+x2+2*x4) = (-x3)],[(x1+3*x2) = (-x4)],[(-x1+x3+2*x4-x5) = (-x5)]]

So I have to do many operations to obtain some vector from this.

Is there some easy way to obtain a simple vector?


Thanks in advance