Difference between revisions of "Unipotent matrix"
From Encyclopedia of Mathematics
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| + | $#C+1 = 10 : ~/encyclopedia/old_files/data/U095/U.0905420 Unipotent matrix | ||
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| − | A matrix group is called unipotent if every matrix in it is unipotent. Any unipotent subgroup of | + | {{TEX|auto}} |
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| + | A square matrix $ A $ | ||
| + | over a ring for which the matrix $ A - I _ {n} $, | ||
| + | where $ n $ | ||
| + | is the order of $ A $, | ||
| + | is nilpotent, i.e. $ ( A - I _ {n} ) ^ {n} = 0 $. | ||
| + | A matrix over a field is unipotent if and only if its [[Characteristic polynomial|characteristic polynomial]] is $ ( x - 1) ^ {n} $. | ||
| + | |||
| + | A matrix group is called unipotent if every matrix in it is unipotent. Any unipotent subgroup of $ \mathop{\rm GL} ( n, F ) $, | ||
| + | where $ F $ | ||
| + | is a field, is conjugate in $ \mathop{\rm GL} ( n, F ) $ | ||
| + | to some subgroup of a special triangular group (Kolchin's theorem). This assertion is also true for unipotent groups over a skew-field, if the characteristic of the latter is either 0 or greater than some $ \gamma ( n) $. | ||
Latest revision as of 08:27, 6 June 2020
A square matrix $ A $
over a ring for which the matrix $ A - I _ {n} $,
where $ n $
is the order of $ A $,
is nilpotent, i.e. $ ( A - I _ {n} ) ^ {n} = 0 $.
A matrix over a field is unipotent if and only if its characteristic polynomial is $ ( x - 1) ^ {n} $.
A matrix group is called unipotent if every matrix in it is unipotent. Any unipotent subgroup of $ \mathop{\rm GL} ( n, F ) $, where $ F $ is a field, is conjugate in $ \mathop{\rm GL} ( n, F ) $ to some subgroup of a special triangular group (Kolchin's theorem). This assertion is also true for unipotent groups over a skew-field, if the characteristic of the latter is either 0 or greater than some $ \gamma ( n) $.
How to Cite This Entry:
Unipotent matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unipotent_matrix&oldid=49081
Unipotent matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unipotent_matrix&oldid=49081
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article