Difference between revisions of "Unimodular transformation"
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| + | A [[Linear transformation|linear transformation]] of a finite-dimensional [[Vector space|vector space]] whose matrix has determinant $ \pm 1 $. | ||
====Comments==== | ====Comments==== | ||
| − | The name "unimodular transformation" is often restricted to mean a linear transformation with determinant | + | The name "unimodular transformation" is often restricted to mean a linear transformation with determinant $ 1 $. |
| + | In the context of a vector space $ V $ | ||
| + | over a field $ k $ | ||
| + | which is the quotient field of an integral domain $ D $, | ||
| + | with a fixed $ k $- | ||
| + | basis $ a _ {1} \dots a _ {n} $ | ||
| + | in $ V $, | ||
| + | a linear transformation is called unimodular if its matrix with respect to $ a _ {1} \dots a _ {n} $ | ||
| + | has entries in $ D $ | ||
| + | and determinant a unit in $ D $. | ||
| + | Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant $ 1 $ | ||
| + | one often calls this the [[Unimodular group|unimodular group]], or, more commonly nowadays, the [[Special linear group|special linear group]]. | ||
Latest revision as of 08:27, 6 June 2020
A linear transformation of a finite-dimensional vector space whose matrix has determinant $ \pm 1 $.
Comments
The name "unimodular transformation" is often restricted to mean a linear transformation with determinant $ 1 $. In the context of a vector space $ V $ over a field $ k $ which is the quotient field of an integral domain $ D $, with a fixed $ k $- basis $ a _ {1} \dots a _ {n} $ in $ V $, a linear transformation is called unimodular if its matrix with respect to $ a _ {1} \dots a _ {n} $ has entries in $ D $ and determinant a unit in $ D $. Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant $ 1 $ one often calls this the unimodular group, or, more commonly nowadays, the special linear group.
How to Cite This Entry:
Unimodular transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_transformation&oldid=14765
Unimodular transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_transformation&oldid=14765
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article