1. ## matrix nilpotencies

Hi everyone. Do you know if any matrix exists with a nilpotency of 0?
A nilpotent matrix N is a square matrix defined as N^k=0 for k>=0....but isn't any square matrix to the zero power the identity matrix....so there is no nilpotent matrix with nilpotency 0??
is my thinking correct?

2. ## Re: matrix nilpotencies

Originally Posted by cp05
....so there is no nilpotent matrix with nilpotency 0?? is my thinking correct?
Yes, you are right.

3. ## Re: matrix nilpotencies

yes. nilpotency is only defined for positive powers of k. A^0 is by definition the identity matrix, for any non-zero matrix A. the zero matrix is a special case, just like the number 0^0 is a special case, although most sources (wolframalpha/mathematica, for example) define it as I (which, oddly enough, is a conflict when one considers 1x1 matrices).