Is N a Zero Matrix or Similar to a Specific Matrix?

[meanyack]

if square of N is zero?

Homework Statement

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]

Homework Equations

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

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.