General linear group

Groups

Group theory
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GL_n(R) or GL(n, R). In Mathematics, the real numbers may be described informally in several different ways

More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French ^[1] Typical notation is GL_n(F) or GL(n, F), or simply GL(n) if the field is understood.

The special linear group, written SL(n, F) or SL_n(F), is the subgroup of GL(n, F) consisting of matrices with determinant =1. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The group GL(n, F) and its subgroups are often called linear groups or matrix groups. These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations The modular group may be realised as a quotient of the special linear group SL(2, Z). In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced

If n ≥ 2, then the group GL(n, F) is not abelian. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

1 General linear group of a vector space
2 In terms of determinants
3 As a Lie group
- 3.1 Real case
- 3.2 Complex case
4 Over finite fields
5 Special linear group
6 Other subgroups
- 6.1 Diagonal subgroups
- 6.2 Classical groups
7 Related groups
- 7.1 Projective linear group
- 7.2 Affine group
8 Infinite general linear group
9 Notes
10 See also
11 Further reading

General linear group of a vector space

If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V, i. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself e. the set of all bijective linear transformations V → V, together with functional composition as group operation. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that If V has finite dimension n, then GL(V) and GL(n, F) are isomorphic. In Mathematics, the dimension of a Vector space V is the cardinality (i In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in The isomorphism is not canonical; it depends on a choice of basis in V. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from Basis vector redirects here For basis vector in the context of crystals see Crystal structure. Given a basis (e₁, . . . , e_n) of V and an automorphism T in GL(V), we have

$Te_k = \sum_{j=1}^n a_{jk} e_j$

for some constants a_jk in F; the matrix corresponding to T is then just the matrix with entries given by the a_jk.

In a similar way, for a commutative ring R the group GL(n, R) may be interpreted as the group of automorphisms of a free R-module M of rank n. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction One can also define GL(M) for any module, but in general this is not isomorphic to GL(n, R) (for any n).

In terms of determinants

Over a field F, a matrix is invertible if and only if its determinant is nonzero. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Therefore an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant.

Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i Therefore GL(n, R) may be defined as the group of matrices whose determinants are units.

Over a non-commutative ring R, determinants are not at all well behaved. In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R). In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring

As a Lie group

Real case

The general linear GL(n,R) over the field of real numbers is a real Lie group of dimension n². In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group To see this, note that the set of all n×n real matrices, M_n(R), forms a real vector space of dimension n². In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The subset GL(n,R) consists of those matrices whose determinant is non-zero. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n The determinant is a polynomial map, and hence GL(n,R) is a open affine subvariety of M_n(R) (a non-empty open subset of M_n(R) in the Zariski topology), and therefore^[2] a smooth manifold of the same dimension. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus.

The Lie algebra of GL(n,R) consists of all n×n real matrices with the commutator serving as the Lie bracket. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

As a manifold, GL(n,R) is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of The identity component, denoted by GL⁺(n, R), consists of the real n×n matrices with positive determinant. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity This is also a Lie group of dimension n²; it has the same Lie algebra as GL(n,R).

The group GL(n,R) is also noncompact. The maximal compact subgroup of GL(n, R) is the orthogonal group O(n), while the maximal compact subgroup of GL⁺(n, R) is the special orthogonal group SO(n). In Mathematics, a maximal compact subgroup K of a Topological group G is a Subgroup K that is a Compact space In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n As for SO(n), the group GL⁺(n, R) is not simply connected (except when n=1), but rather has a fundamental group isomorphic to Z for n=2 or Z₂ for n>2. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

Complex case

The general linear GL(n,C) over the field of complex numbers is a complex Lie group of complex dimension n². Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group As a real Lie group it has dimension 2n². The set of all real matrices forms a real Lie subgroup.

The Lie algebra corresponding to GL(n,C) consists of all n×n complex matrices with the commutator serving as the Lie bracket. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

Unlike the real case, GL(n,C) is connected. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of This follows, in part, since the multiplicative group of complex numbers C^× is connected. The group manifold GL(n,C) is not compact; rather its maximal compact subgroup is the unitary group U(n). In Mathematics, a maximal compact subgroup K of a Topological group G is a Subgroup K that is a Compact space In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices As for U(n), the group manifold GL(n,C) is not simply connected but has a fundamental group isomorphic to Z. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

Over finite fields

If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements When p is prime, GL(n, p) is the outer automorphism group of the group Z_pⁿ, and also the automorphism group, because Z_pⁿ is Abelian, so the inner automorphism group is trivial. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G

The order of GL(n, q) is:

(qⁿ - 1)(qⁿ - q)(qⁿ - q²) … (qⁿ - q^n-1)

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the k^th column can be any vector not in the linear span of the first k-1 columns. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector

For example, GL(3,2) has order (8-1)(8-2)(8-4)=168. It is the automorphism group of the Fano plane and of the group Z₂³. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. In Mathematics, a Grassmannian is a space which parameterizes all Linear subspaces of a Vector space V of a given Dimension. This requires only finding the order of the stabilizer subgroup of one such subspace (described on that page in block matrix form), and dividing into the formula just given, by the orbit-stabilizer theorem. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In the mathematical discipline of Matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. In Mathematics, Schubert calculus is a branch of Algebraic geometry introduced in the Nineteenth century by Hermann Schubert, in order to solve In Mathematics, in the area of Combinatorics and Special functions a q -analog is roughly speaking a theorem or identity for a ''q''-series In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing This was one of the clues leading to the Weil conjectures. In Mathematics, the Weil conjectures, which had become theorems by 1974 were some highly-influential proposals from the late 1940s by André Weil on the

Special linear group

Main article: Special linear group

The special linear group, SL(n, F), is the group of all matrices with determinant 1. In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. SL(n, F) is a normal subgroup of GL(n, F). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

If we write F^× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism

det: GL(n, F) → F^×. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

The kernel of the map is just the special linear group. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism By the first isomorphism theorem we see that GL(n,F)/SL(n,F) is isomorphic to F^×. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In fact, GL(n, F) can be written as a semidirect product of SL(n, F) by F^×:

GL(n, F) = SL(n, F) ⋊ F^×

When F is R or C, SL(n) is a Lie subgroup of GL(n) of dimension n² − 1. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, a Subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth The Lie algebra of SL(n) consists of all n×n matrices over F with vanishing trace. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal The Lie bracket is given by the commutator. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of Rⁿ. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which

The group SL(n, C) is simply connected while SL(n, R) is not. SL(n, R) has the same fundamental group as GL⁺(n, R), that is, Z for n=2 and Z₂ for n>2.

Other subgroups

Diagonal subgroups

The set of all invertible diagonal matrices forms a subgroup of GL(n, F) isomorphic to (F^×)ⁿ. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In fields like R and C, these correspond to rescaling the space; the so called dilations and contractions.

A scalar matrix is a diagonal matrix which is a constant times the identity matrix. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main The set of all nonzero scalar matrices forms a subgroup of GL(n, F) isomorphic to F^× . This group is the center of GL(n, F). In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In particular, it is a normal, abelian subgroup.

The center of SL(n, F) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of nth roots of unity in the field F. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power

Classical groups

The so-called classical groups are subgroups of GL(V) which preserve some sort of bilinear form on a vector space V. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where These include the

orthogonal group, O(V), which preserves a non-degenerate quadratic form on V,
symplectic group, Sp(V), which preserves a symplectic form on V (a non-degenerate alternating form),
unitary group, U(V), which, when F = C, preserves a non-degenerate hermitian form on V. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, the name symplectic group can refer to two different but closely related types of mathematical groups. In Mathematics, a symplectic vector space is a Vector space V equipped with a Nondegenerate, Skew-symmetric, Bilinear form In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear

These groups provide important examples of Lie groups.

Related groups

Projective linear group

The projective linear group PGL(n, F) and the projective special linear group PSL(n,F) are the quotients of GL(n,F) and SL(n,F) by their centers (which consists of some multiples of the identity matrix). In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

Affine group

Main article: Affine group

The affine group Aff(n,F) is an extension of GL(n,F) by the group of translations in Fⁿ. In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group. It can be written as a semidirect product:

Aff(n, F) = GL(n, F) ⋉ Fⁿ

where GL(n, F) acts on Fⁿ in the natural manner. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space Fⁿ. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.

One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, SL(n, F) ⋉ Fⁿ, and the [[Poincaré group] is the affine group associated to the Lorentz group, O(1,3,F) ⋉ Fⁿ. In the mathematical study of Transformation groups the special affine group is the group of Affine transformations of a fixed Affine space In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting

Infinite general linear group

The infinite general linear group or stable general linear group is the direct limit of the inclusions $\operatorname{GL}(n,F) \to \operatorname{GL}(n+1,F)$ as the upper left block matrix. In Mathematics, a direct limit of groups is the Direct limit of a direct system of groups In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In the mathematical discipline of Matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices It is denoted by either $\operatorname{GL}(F)$ or $\operatorname{GL}(\infty,F)$ , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.

It is used in algebraic K-theory to define K₁, and over the reals has a well-understood topology, thanks to Bott periodicity. In Mathematics, algebraic K-theory is an advanced part of Homological algebra concerned with defining and applying a sequence K n In Mathematics, algebraic K-theory is an advanced part of Homological algebra concerned with defining and applying a sequence K n In Mathematics, the Bott periodicity theorem is a result from Homotopy theory discovered by Raoul Bott during the latter part of the 1950s which proved

Notes

^ Here rings are assumed to be associative and unital. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real
^ Since the Zariski topology is coarser than the metric topology; equivalently, polynomial maps are continuous. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

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