General linear group

Centre of the general linear group

Let . The centre of the general linear group consists of all homotheties, i.e. multiples of identity, and is isomorphic to the multiplicative group of the underlying field . group

Proof

It is clear that scalar transformations form a group isomorphic to and that . Now let and assume there exists a nonzero such that not an eigenvector of . Then is linearly independent from , and there exists some vector basis with and . Let and . Define linear maps such that in the basis

and . Then

implying , a contradiction. Hence every must be an eigenvector of , so .

Properties

• Scalar transformation criterion

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