if square of N is zero???

[meanyack]

1. The problem statement, all variables and given/known data
Let N be 2x2 matrix such that N2=0. How can we prove either N=0 or N is similar over C to [0 0; 1 0]

2. Relevant equations

Two matrix is to be similar if A=P-1BP for invertible transformation matrix P



3. The attempt at a solution
I tried to multiply N by itself but I got square of indices and some complex variables so I think that's not working.

[Office_Shredder]

Start by noting that if N2 = 0 then N is non-invertible. What else can you conclude?

[meanyack]

use of det(N)=ad-bc works while finding eigenvalue, thanks.

[HallsofIvy]

N2= 0 means that N2v= 0= 0v for all v. 0 is a double eigenvalue. N2v= N(Nv)= 0.
Either Nv= 0 or Nv is in the null space of N.