A square matrix A is said to be unipotent if A-I, where I is an identity matrix is a nilpotent matrix (defined by the property that A^n is the zero matrix for some positive integer matrix power n. The corresponding identity, (A-I)^k=0 for some integer k allows this definition to be generalized to other types of algebraic systems.

An example of a unipotent matrix is a square matrix whose entries below the diagonal are zero and its entries on the diagonal are one. An explicit example of a unipotent matrix is given by

 [1 1 0 0; 0 1 1 0; 0 0 1 1; 0 0 0 1].

One feature of a unipotent matrix is that its matrix powers A^n have entries which grow like a polynomial in n.

A semisimple element x of a group G is unipotent if F^*(C_G(x)) is a p-group, where F^* is the generalized fitting subgroup.

See also

Fitting Subgroup, Iwasawa Decomposition, Nilpotent Matrix, p-Group, Semisimple Element

This entry contributed by Todd Rowland

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Rowland, Todd. "Unipotent." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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