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Wikipedia / General linear group

In mathematics, the general linear group 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, with the identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

(the set of real numbers.

over any field (such as the complex numbers), or a ring (such as the ring of integers), again with matrix multiplication as the group operation. if the field is understood.

More generally still, the general linear group of a vector space is the automorphism group, not necessarily written as matrices.

The special linear group, is the subgroup consisting of matrices with a determinant of 1.

and its subgroups are often called linear groups or matrix groups is a linear group but not a matrix group). 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.

is not abelian.

General linear group of a vector space

is a vector space, is the group of all automorphisms, i.e. the set of all bijective linear transformations has finite dimension are isomorphic. The isomorphism is not canonical; it depends on a choice of basis, we have then for every basis vector ei that

.

may be interpreted as the group of automorphisms of a free. One can also define GL(M).

In terms of determinants

, a matrix is invertible if and only if its determinant is as the group of matrices with nonzero determinant.

Over a commutative ring is invertible if and only if its determinant is a unit may be defined as the group of matrices whose determinants are units.

may be defined as the unit group of the matrix ring.

As a Lie group/algebra

Real case

over the field of real numbers is a real Lie group, forms a real vector space consists of those matrices whose determinant is non-zero. The determinant is a polynomial is an open affine subvariety (a non-empty open subset in the Zariski topology), and therefore a smooth manifold of the same dimension.

The Lie algebra real matrices with the commutator serving as the Lie bracket.

is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The identity component.

The polar decomposition with the set of positive-definite symmetric matrices. Because the latter is contractible, the fundamental group.

is noncompact. “The” maximal compact subgroup is the orthogonal group is the special orthogonal group is not simply connected, but rather has a fundamental group.

Complex case

The general linear group over the field of complex numbers, is a complex Lie group. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions

  1. ,

-dimensional matrices that preserve a linear complex structure.

The Lie algebra complex matrices with the commutator serving as the Lie bracket.

is connected is not compact; rather its maximal compact subgroup is the unitary group is not simply connected but has a fundamental group.

Over finite fields

is a finite field. When p is the outer automorphism group, and also the automorphism is abelian, so the inner automorphism group is trivial.

is:

th column can be any vector not in the linear span columns. In q-analog.

For example, has order . It is the automorphism group of the Fano plane. This group is also isomorphic to .

More generally, one can count points of Grassmannian. 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.

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.

goes to 0! ) we see that it is the order of the symmetric group (see Lorscheid's article). In the philosophy of the field with one element, one thus interprets the symmetric group.

History

, 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.

Special linear group

The special linear group, is the group of all matrices with determinant 1. These matrices are special in that they lie on a subvariety: they satisfy a polynomial equation (as the determinant is a polynomial in the entries). 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.

for the multiplicative group excluding 0), then the determinant is a group homomorphism

that is surjective and its kernel is a normal subgroup, and by the first isomorphism theorem is isomorphic can be written as a semidirect product:

  1. .

The special linear group is also the derived group (for a field or a division ring is not the field with two elements.

is a Lie subgroup. The Lie algebra with vanishing trace. The Lie bracket is given by the commutator.

can be characterized as the group of volume and orientation-preserving.

.

Other subgroups

Diagonal subgroups

The set of all invertible diagonal matrices, 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. This group is the center. In particular, it is a normal, abelian subgroup.

th roots of unity.

Classical groups

The so-called classical groups which preserve some sort of bilinear form. These include the

These groups provide important examples of Lie groups.

Related groups and monoids

Projective linear group

The projective linear group and the projective special linear group are the quotients by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.

Affine group

The affine group is an extension. It can be written as a semidirect product:

in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space.

One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group, and the Poincaré group is the affine group associated to the Lorentz group.

General semilinear group

The general semilinear group is the group of all invertible semilinear transformations. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a field automorphism under scalar multiplication”. It can be written as a semidirect product:

is the Galois group (over its prime field by the Galois action on the entries.

is that the associated projective semilinear group, is the collineation group of projective space, and thus semilinear maps are of interest in projective geometry.

Full linear monoid

If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually a regular semigroup.

Infinite general linear group

The infinite general linear group or stable general linear group is the direct limit as block diagonal matrices, 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.

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.

See also

Notes

References