General linear group

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In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. (This is indeed a group because the product of two invertible matrices is again invertible, as is the inverse of one.) If the field is clear from context we sometimes write GL(n), or GLn.

The special linear group, written SL(n, F) or SL(n), is the subgroup of GL(n, F) consisting of matrices with determinant 1.

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.

If n ≥ 2, then the group GL(n, F) is not abelian.

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 VV, together with functional composition as group operation. If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism.

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, Mn(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 Mn(R) and therefore smooth manifold of the same dimension.

The Lie algebra of GL(n,R) consists of all n×n real matrices with the commutator serving as the Lie bracket.

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).

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.

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.

The Lie algebra corresponding to GL(n,C) consists of all n×n complex matrices with the commutator serving as the Lie bracket.

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.

Over finite fields

If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). GL(n, q) is the outer automorphism group of the group Zpn, and also the automorphism group, because Zpn is Abelian, so the inner automorphism group is trivial.

The order of GL(n, q) is:

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

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero column; the second column can be anything but the multiples of the first column, etc.

For example GL(3,2) has order 168. It is the automorphism group of the Fano plane and of the group Z23.

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.

The connection between these formulae, and the Betti numbers of complex Grassmannians, was one of the clues leading to the Weil conjectures.

Special linear group

The special linear group, SL(n, F), is the group of all matrices with determinant 1. That this forms a group follows from the rule of multiplication of determinants. SL(n, F) is a normal subgroup of GL(n, F).

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

det: GL(n, F) → F×.

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×:

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. 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.

The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of Rn.

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 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. The set of all nonzero scalar matrices, sometimes denoted Z(n, F), 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.

The center of SL(n, F), denoted SZ(n, F), is simply the set of all scalar matrices with unit determinant. Note that SZ(n, C) is isomorphic to the nth roots of unity.

Classical groups

The so-called classical groups are subgroups of GL(V) which preserve some sort of inner product on V. These include the

These groups provide important examples of Lie groups.

See also

de:Allgemeine lineare Gruppe es:Grupo general lineal fr:Groupe général linéaire zh:一般线性群