In mathematics, the general linear group of degree n {\displaystyle n} is the set of n × n {\displaystyle n\times n} invertible matrices, together with...

especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced...

groups). A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GLd (K)...

with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the...

translations, and the affine group of A can be described concretely as the semidirect product of V by GL(V), the general linear group of V: Aff ⁡ ( V ) = V ⋊...

mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to...

orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...

{\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ⁡ ( n ) ⊂ U ⁡ ( n ) ⊂ GL ⁡ ( n , C ) . {\displaystyle \operatorname...

In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)...

automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its...

abelian group and its automorphism group is a general linear group, so very well understood. The map from the automorphism group of G into this general linear...

of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ⁡ (...

similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices...

group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix...

simply a group homomorphism ρ : G → G L ( n , R ) {\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )} from the group to the general linear group. This...

unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices...

(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) The Lie algebra of the general linear group GL(n...

For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same...

scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: GL n ≠ SL...

variety. A general linear group is an example of a linear algebraic group, an affine variety that has a structure of a group in such a way the group operations...

construction of the unitary group from the general linear group. The unitary group arises as follows: the general linear group over the complex numbers has...

mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field...

theory of Coxeter groups, the symmetric group is the Coxeter group of type An and occurs as the Weyl group of the general linear group. In combinatorics...

multiplication is R {\displaystyle \mathbb {R} } -linear (but not H {\displaystyle \mathbb {H} } -linear) we have another embedding of H × {\displaystyle...

bases is in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(R)} - the general linear group of R (in simple terms this means that all the entries of T are in...

all ordered bases, or frames, for E x {\displaystyle E_{x}} . The general linear group acts naturally on F ( E ) {\displaystyle F(E)} via a change of basis...

affine group, which has the general linear group of degree n {\displaystyle n} as subgroup and is itself a subgroup of the general linear group of degree...

a linear space with a basis. Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The...

the general linear group GLn (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over...

_{p}),} where G L {\displaystyle \mathrm {GL} } is the appropriate general linear group. This is easily shown to have order | Aut ⁡ ( P ) | = ( p n − 1 )...