matrix group

matrix group

English

Pronunciation

• Rhymes: -uːp

Noun

matrix group (plural matrix groups)

1. (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

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 GL_n(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

• (group of invertible matrices): linear group

See also

• unitary group