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.
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 GLn(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.
The special linear group, written SL(n, F) or SLn(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1.
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.
Therefore, an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant.
In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R).
The determinant is a polynomial map, and hence GL(n, R) is an open affine subvariety of Mn(R) (a non-empty open subset of Mn(R) in the Zariski topology), and therefore[2] a smooth manifold of the same dimension.
consists of all n×n real matrices with the commutator serving as the Lie bracket.
The identity component, denoted by GL+(n, R), consists of the real n×n matrices with positive determinant.
Similarly, it shows that there is a homeomorphism between GL+(n, R) and the Cartesian product of SO(n) with the set of positive-definite symmetric matrices.
Complex n-dimensional matrices can be characterized as real 2n-dimensional matrices that preserve a linear complex structure — concretely, that commute with a matrix J such that J2 = −I, where J corresponds to multiplying by the imaginary unit i.
This follows, in part, since the multiplicative group of complex numbers C∗ is connected.
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 and dividing into the formula just given, by the orbit-stabilizer theorem.
The general linear group over a prime field, GL(ν, p), was constructed and its order computed by Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the Galois group of the general equation of order pν.
In fact, GL(n, F) can be written as a semidirect product: The special linear group is also the derived group (also known as commutator subgroup) of the GL(n, F) (for a field or a division ring F) provided that
The Lie algebra of SL(n, F) consists of all n×n matrices over F with vanishing trace.
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.
The set of all nonzero scalar matrices forms a subgroup of GL(n, F) isomorphic to F×.
It can be written as a semidirect product: where GL(n, F) acts on Fn in the natural manner.
[7] The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix.
[12] It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity.
It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible – see Kuiper's theorem.
