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?
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).