What is the center of SL(n,C)?

[cmj1988]

What is the center of SL(n,C)?

I understand that the center of a group is where all elements commute with the group G. So I figure that I should come up with a case in which matricies commute. I remember a few facts from Linear Algebra:

Fact 1: Simultaneously diagonalizeanle matricies lend itself to commutivity
Fact 2: Matrix multiplication is associative

So given an M in SL(n,C) and an A, B in GL(n,C):
D₁=MAM^-1
D₂=MBM^-1

So, AB=M^-1D₁MM^-1D₂M=M^-1D₁D₂M

We know that diagonal matricies are commutative

M^-1D₂D₁M

Invoking associativity

BA

I'm not sure if I actually answered the question.

 

[HallsofIvy]

As you said at the beginning, "the center of a group is where all elements commute with the group G". "Simultaneously diagonalizable" matrices commute with each other but not, generally, with other matrices. Also, matrices A and B may be simultaneously diagonalizable, and matrices C and D may be simultaneously diagonalizable, but A and B not simultaneously diagaonalizabel with C and D. Which pair would you take as center?

Instead of thinking about "diagonalizable" matrices, look at diagonal matrices.

 

[cmj1988]

So all diagonal matrices?