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Jim Kata
- 204
- 10
Is the center of GL(n) the set diagonal martices? It's easy to prove that ADA^-1 = D if D is diagonal, but if ABA^-1 = B is true that B is diagonal.
GL(n) stands for the general linear group, which consists of all invertible n-by-n matrices. In other words, it is the group of all non-singular matrices of size n.
Center diagonal matrices are square matrices where all elements outside the main diagonal (the diagonal from the top left corner to the bottom right corner) are zero. The elements on the main diagonal can be any non-zero numbers.
To prove that all center diagonal matrices are in the center of GL(n), we need to show that they commute with all other matrices in GL(n). This can be done by multiplying a center diagonal matrix with an arbitrary matrix in GL(n) and showing that the result is the same when the order of multiplication is reversed.
Proving that GL(n) center diagonal matrices are in the center is important because it helps us understand the structure of the general linear group. It also has applications in fields such as linear algebra, group theory, and physics.
Yes, an example of a center diagonal matrix in GL(n) is the identity matrix, where all elements on the main diagonal are 1 and all other elements are 0. This matrix commutes with all other matrices in GL(n) and is therefore in the center.