matrix group
English
[edit]Pronunciation
[edit]- Rhymes: -uːp
Noun
[edit]matrix group (plural matrix groups)
- (group theory) Any group of invertible matrices over a specified field, with the group operation of matrix multiplication.
- 1976, S. Califano, Vibrational States, Wiley, page 112:
- The properties of matrix groups are of great importance for representation theory.
- 2011, László Babai, Finite Groups and Complexity Theory: From Leningrad to St Petersburg via Las Vegas, Alexander Kulikov, Nikolay Vereshchagin (editors), Computer Science – Theory and Applications: 6th International Computer Symposium, Springer, Lecture Notes in Computer Science 6651, page 162,
- This paper is a personal account of the author's journey through the evolution of some of these interconnections, culminating in recent definitive results on the matrix group membership problem.
- 2014, Peter Sarnak, Notes on thin matrix groups, Emmanuel Breuillard, Hee Oh (editors), Thin Groups and Superstrong Approximation, Cambridge University Press, page 343,
- Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given.
Usage notes
[edit]A group isomorphic to a matrix group is called a linear group and is said to be linear. In mathematical terms, for any linear group G, there exist a field K, an integer d and an injective homomorphism from G to (for some n) the general linear group GLn(K) that is a faithful linear representation of dimension d over K. G can be said to be linear of degree d over K.
Synonyms
[edit]- (group of invertible matrices): linear group