Is a 2x2 Complex Matrix with N^2=0 Always Similar to a Specific Form?

[Alupsaiu]

Suppose N is a 2x2 complex matrix such that N^2=0. Prove that either N=0 or N is similar over C to the matrix

00
10Sorry, I don't know how else to write the matrix in the post. Any help would be greatly appreciated, thank you.

 

[Petr Mugver]

If N is not zero, then it exists a vector v such that Nv is not zero. Using the fact that N^2=0 it is easy to proove that v and Nv are linearly independent (try it), so they are a basis of C^2. Try writing N with respect to this basis.

 

[Alupsaiu]

Ok, so it was easy to show linear independence, then I found the matrix which had in the first column the coordinates of Nv, and in the second column 0 since N^2=0.
So,

x1 0
x2 0, where x1 and x2 are the coordinates of Nv in basis (v, Nv). Would x1 actually turn out to be 0?


But, where to go from here? In general I'm a bit confused on how to show a matrix is similar, or must be similar to another matrix. Again, thank you for the help.

 

[zcd]

If {v, Nv} is your basis, then writing N as a matrix with respect to this basis would be of the form [N(v) N(Nv)] where N(v) and N(Nv) are column vectors. If you use the fact that N^2= 0, the rest will follow.

As for showing similarity, recall that matrices are self similar even after change of basis.