NationMaster - Encyclopedia: General linear group

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. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... This article gives an overview of the various ways to multiply matrices. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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). Please refer to Real vs. ...

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.^[1] Typical notation is GL_n(F) or GL(n, F), or simply GL(n) if the field is understood. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

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 G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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 abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of...

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. The modular group may be realised as a quotient of the special linear group SL(2, Z). Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

If n ≥ 2, then the group GL(n, F) is not abelian. Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in...

General linear group of a vector space

If V is a vector space over the field F, then we write GL(V) or Aut(V) for the group of all automorphisms of V, i.e. the set of all bijective linear transformations V → V, together with functional composition as group operation. If V has finite dimension n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Given a basis (e₁, ..., e_n) of V and an automorphism T in GL(V), we have In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... 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 a way that respects the given group operations. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

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 module having a free basis. ...

In terms of determinants

Over a field F, a matrix is invertible if and only if its determinant is nonzero. 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. Therefore GL(n, R) may be defined as the group of matrices whose determimants are units. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

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 M(n, R). In mathematics, a unit in a (unital) ring R is an invertible element of R, i. ...

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. To see this, note that the set of all n×n real matrices, M_n(R), forms a real vector space of dimension n². The subset GL(n,R) consists of those matrices whose determinant is non-zero. The determinant is a continuous (even polynomial) map, and hence GL(n,R) is a non-empty open subset of M_n(R) and therefore smooth manifold of the same dimension. In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

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. ... In mathematics, the commutator gives an indication of how poorly 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. The identity component, denoted by GL⁺(n, R), consists of the real n×n matrices with positive determinant. This is also a Lie group of dimension n²; it has the same Lie algebra as GL(n,R). Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ... Connected components come up in topology and in graph theory, two related branches of mathematics. ... In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ...

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). 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 mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... 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. As a real Lie group it has dimension 2n². The set of all real matrices forms a real Lie subgroup. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

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. ... In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

Unlike the real case, GL(n,C) is connected. 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). As for U(n), the group manifold GL(n,C) is not simply connected but has a fundamental group isomorphic to Z. Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

As an algebraic variety

The general linear group of degree n over a field F may be regarded as an open subvariety of affine n²-space over F.

Over finite fields

If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). 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 abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by xa. ...

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 space is the intersection of all subspaces containing that set. ...

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. A finite geometry is any geometric system that has only a finite number of points. ...

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one (described on that page in block matrix form), and divide into the formula just given, by the orbit-stabilizer theorem. In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... This article is about mathematical concept. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In mathematics, groups are often used to describe symmetries of objects. ...

The connection between these formulae, and the Betti numbers of complex Grassmannians, was one of the clues leading to the Weil conjectures. In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ...

Special linear group

The special linear group, SL(n, F), is the group of all matrices with determinant 1. 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 algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

If we write F^× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

The kernel of the map is just the special linear group. By the first isomorphism theorem we see that GL(n,F)/SL(n,F) is isomorphic to F^×. In fact, GL(n, F) can be written as a semidirect product of SL(n, F) by F^×: 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 fails to be injective. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...

When F is R or C, SL(n) is a Lie subgroup of GL(n) of dimension n² − 1. The Lie algebra of SL(n) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutator. In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... 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 diagonal from the upper left to the lower right) of A, i. ... In mathematics, the commutator gives an indication of how poorly 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. Volume, also called capacity, is a quantification of how much space an object occupies. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...

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.

Simple groups

The projective special linear groups PGL_n(F) over fields F are simple for n≥ 2 with two exceptions: the groups PGL₂(F₃) and PGL₂(F₂) with n=2 and F a finite field of order 2 or 3 are solvable.

Other subgroups

Diagonal subgroups

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

A scalar matrix is a diagonal matrix which is a constant times the identity matrix. 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 particular, it is a normal, abelian subgroup. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of...

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, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...

Classical groups

The so-called classical groups are subgroups of GL(V) which preserve some sort of bilinear form on a vector space V. These include the In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ...

These groups provide important examples of Lie groups. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ... 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 ω called the symplectic form. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...

Notes

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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