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.
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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 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.
Over R and C
If the field F is R (the real numbers) or C (the complex numbers), then GL(n) is a real/complex Lie group of real/complex dimension n2. The reason is as follows: GL(n) consists of those matrices whose determinant is non-zero, the determinant is a continuous (even polynomial) map, and hence GL(n) is a non-empty open subset of the manifold of all n-by-n matrices, which has dimension n2.
The Lie algebra corresponding to GL(n) consists of all n×n matrices over F, with the commutator serving as the Lie bracket.
While GL(n, C) is connected, GL(n, R) has two connected components: the matrices with positive determinant and the ones with negative determinant. The real n-by-n matrices with positive determinant form a subgroup of GL(n, R) denoted by GL+(n, R). This is also a Lie group of real dimension n2 and it has the same Lie algebra as GL(n, R). GL+(n,R) is connected.
Neither GL(n, C) nor GL+(n, R) is simply connected (expect, in the real case, when n=1). The group manifold GL(n, C) has a fundamental group isomorphic to Z while GL+(n, R) has a fundamental group isomorphic to Z for n=2 and Z2 for n>2.
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 a finite group with
- (qn - 1)(qn - q)(qn - q2) … (qn - qn-1)
elements. 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.
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
- orthogonal group, O(V), which preserves a symmetric bilinear form on V,
- symplectic group, Sp(V), which preserves a skew-symmetric bilinear form on V,
- unitary group, U(V), which preserves a hermitian form on V (when F = C).
These groups provide important examples of Lie groups.