1. complex space

how do you show that the center of a general linear matrix over complex space is isomorphic to complex numbers excluding 0?

thanks

2. The center in this case is $H=\{\lambda I:\lambda\in \mathbb{C}-\{0\}\}$ .

Prove that $f:H \to \mathbb{C}-\{0\},\;f(\lambda I)=\lambda$ is an isomorphism.

Fernando Revilla

3. in this case, i know how to prove that f(λI) = λ is isomorphic...which would be by showing that it is homo, the kernel is the identity matrix and it is onto right?

but how did you manage to find out that the center can be denoted as λI and to define the function? I dont know how to do that..

thanks

4. Originally Posted by alexandrabel90
in this case, i know how to prove that f(λI) = λ is isomorphic...which would be by showing that it is homo, the kernel is the identity matrix and it is onto right?

Right.

but how did you manage to find out that the center can be denoted as λI and to define the function? I dont know how to do that..

By a well known theorem, $A\in \mathbb{K}^{n\times n}$ conmutes with all $X\in \mathbb{K}^{n\times n}$ iff $A$ is a scalar matrix.

Fernando Revilla