Re: Center of a matrix group

Let a matrix in the center. First we will try to get information about the fact that for all diagonal matrix . We must have and if . If we fix we must have since we can choose a matrix such that . Hence is a diagonal matrix: . Now if you take with you will get that for all and .

Re: Center of a matrix group

Any element in the center of the group must commute with an arbitrary matrix. The idea is to use very simple matrices, multiply on both sides, and get conditions that your matrix would have to satisfy so that it would commute with your simple matrix.

These "simple matrices" will be elementary matrices; that is, matrices that correspond to elementary column (when multiplying on the right) or row (when multiplying from the left) operations. For example, if is in the center of the group, then we should have

What does this tell us about ?

(Note: That matrix is supposed to be , where represents the identity matrix, and the matrix with a 1 in position (i,j) and 0's elsewhere.)