2 By 2 Matrices Abstract Algebra General Linear 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.

2 by 2 matrices abstract algebra general linear group. The objective of this paper is to derive formulae for the generalized drazin inverse of a block matrix in a banach algebra. The set of all invertible n n matrices with entries in r is called the general linear group of degree n over the real numbers and is denoted by gl n r. Let g 1 and g 2 be groups. 22 invertible upper triangular matrices form a subgroup of general linear group.

Definition the general linear group is defined as the group of invertible matrices with entries from the field of real numbers and with the group operation being matrix multiplication. If n 2 then the group gln f is not abelian. M2 r is also a four dimensional vector space so it is considered an associative algebra. The group gln f and its subgroups are often called linear groups or matrix groups the abstract group glv is a linear group but not a matrix group.

The proof that gl2z3 is a group under matrix multiplication follows the proof in the last example. Let mathsfb1 be the polynomial ring mathbbcapm1b with the structure of a complex hopf algebra induced from its interpretation as the algebra of regular functions on the affine linear algebraic group of complex invertible upper triangular 2 by 2 matrices of the form left beginsmallmatrix ab01 endsmallmatrixright. Abstract algebra dummit foote. The general linear group is written as glnf where f is the field used for the matrix elements.

Thus recent results on the drazin inverse of block matrices and block operator matrices are extended to a more general setting. Here we introduce matrix groups with an emphasis on the general linear group and special linear group. It is a particular case of general linear group of degree two general linear group over reals and more generally of a general linear group. For example 2 1 1 2 1 1 2 1 1 0 2 0.

The operation is matrix multiplication but note that all the arithmetic is performed in z3. We prove that the universal invariant. In terms of abstract algebra m2 r with the associated addition and multiplication operations forms a ring and gl2 r is its group of units. Gl2z3 denotes the set of 22 invertible matrices with entries in z3.