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.

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).

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.

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.

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).

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, the general linear group of V, written GL(V) or Aut(V), is 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

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. 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. 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 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).

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 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.

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). 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.

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.

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.

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 such subspace (described on that page in block matrix form), and dividing into the formula just given, by the orbit-stabilizer theorem.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. This 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. 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).

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 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) 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.

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

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.

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).

Affine group

The affine group Aff(n,F) is an extension of GL(n,F) by the group of translations in Fⁿ. 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. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space Fⁿ.

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. 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.

Notes

^ Here rings are assumed to be associative and unital.
^ Since the Zariski topology is coarser than the metric topology; equivalently, polynomial maps are continuous.