Thread: Problem 38
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Old September 24th, 2007, 07:18 PM
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Default Problem 38
Show the center of the general linear group are the proportional diagnol matrices.
Meaning, Z(\mbox{GL}_n(\mathbb{R})) = \{ k I| k\in \mathbb{R}^{*} \mbox{ and }I \mbox{ identity matrix} \}.

I try to explain this so it makes it more elementary. The set \mbox{GL}_n(\mathbb{R}) is the set of all n\times n invertible matrices having \mathbb{R} as their entries. This set is called the "general linear group". The "center" of this set are all the matrices which commute with everything else. So you need to show if a matrix commute with all invertible matrices then it must be a proportional diagnol matrix.
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