Matrixgruppen

Matrixgruppen, or matrix groups, are groups whose elements are invertible matrices under multiplication. They are constructed from n×n matrices with entries in a field F (or, more generally, a commutative ring) and form a group under matrix multiplication. The most fundamental example is the general linear group GL(n, F) = {A ∈ M_n(F) | det(A) ≠ 0}, the set of all invertible n×n matrices.

Subgroups defined by determinant or form-preserving conditions include the special linear group SL(n, F) = {A | det(A) = 1}, the orthogonal group O(n) = {A | A^T A = I}, the unitary group U(n) = {A | A* A = I}, and the symplectic group Sp(2n, F). For finite fields F_q, GL(n, q) is a finite group with order (q^n − 1)(q^n − q)…(q^n − q^{n−1}). These groups can be studied over various fields and rings, producing a wide range of algebraic and geometric objects.

Matrix groups often carry rich algebraic and geometric structures. Over the real or complex numbers, GL(n, F) and many of its subgroups have the structure of Lie groups, with dimensions such as n^2 for GL(n, F). The determinant map det: GL(n, F) → F^× shows that SL(n, F) is its kernel, reflecting a common structural decomposition. Matrix groups act naturally on vector spaces, underpin representations, and play central roles in physics, geometry, computer graphics, and data analysis.