The general linear group is the group of invertible linear maps from a vector space to itself, with composition as its operation.
A monomial matrix has exactly one nonzero entry in each row and column. It factors as an invertible diagonal matrix times a permutation matrix. Such matrices permute the coordinate axes and normalize the group of invertible diagonal matrices; over very small fields that normalizer can be larger if the diagonal subgroup fails to distinguish the axes.
The integral general linear group consists of integer matrices with determinant . These are exactly the integer matrices whose inverses also have integer entries, so they are the automorphisms of the free abelian group .
A matrix group is a group represented by invertible matrices, equivalently a subgroup of a general linear group.
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The General Linear Group, denoted as \( \text{GL}(n, F) \), is a fundamental concept in linear algebra and group theory. It consists of all invertible \( n \times n \) matrices with entries from a field \( F \).
Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.