# Center of a matrix group

• Jun 25th 2011, 03:04 AM
sashikanth
Center of a matrix group
The question is for me to find the center of a group comprising of non singular square matrices of order n.

So if any member of the center is A, and G is any matrix in the group, we need AG = GA --> A = G-1 * A * G. (Where G-1 = G inverse)

I can intuitively say that only elements of the form A = nI satisfy this equation, but how do I prove it?

• Jun 25th 2011, 03:26 AM
girdav
Re: Center of a matrix group
Let $\displaystyle A$ a matrix in the center. First we will try to get information about the fact that $\displaystyle AD=DA$ for all diagonal matrix $\displaystyle D$. We must have $\displaystyle (AD)_{ij} = \alpha_ja_{ij}$ and $\displaystyle (DA)_{ij} = a_{ij}\alpha_i$ if $\displaystyle D = \mathrm{diag}(\alpha_1,\ldots,\alpha_n)$. If we fix $\displaystyle i\neq j$ we must have $\displaystyle a_{ij} =0$ since we can choose a matrix $\displaystyle D$ such that $\displaystyle \alpha_i\neq \alpha_j$. Hence $\displaystyle A$ is a diagonal matrix: $\displaystyle A =\mathrm{diag}(a_1,\ldots,a_n)$. Now if you take $\displaystyle G$ with $\displaystyle g_{ij} = i$ you will get that $\displaystyle a_i=a_j$ for all $\displaystyle i$ and $\displaystyle j$.
• Jun 25th 2011, 03:29 AM
topspin1617
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 $\displaystyle A$ is in the center of the group, then we should have

$\displaystyle \begin{pmatrix}-1&0&\cdots&\\0&1&0&\cdots\\0&0&1&\cdots\\\cdots&\c dots&&\end{pmatrix}A=A\begin{pmatrix}-1&0&\cdots&\\0&1&0&\cdots\\0&0&1&\cdots\\\cdots&\c dots&&\end{pmatrix}$

What does this tell us about $\displaystyle A$?

(Note: That matrix is supposed to be $\displaystyle I_n-2E_{1,1}$, where $\displaystyle I_n$ represents the $\displaystyle n\times n$ identity matrix, and $\displaystyle E_{i,j}$ the matrix with a 1 in position (i,j) and 0's elsewhere.)