Jordan Decomposition
Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gugs of commuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan–Chevalley decomposition.
There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group.
Read more about this topic: Unipotent
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