Matrix Representations

Core Idea

Once a basis for V is chosen, a representation ρ: G → GL(V) becomes a map G → GL_n(F), assigning an n×n invertible matrix to each group element. Different basis choices yield different matrix representations of the same abstract representation. The passage from abstract linear maps to concrete matrices makes computation possible but introduces basis-dependence that must be carefully managed.

Explainer

An abstract representation ρ: G → GL(V) becomes a matrix representation once you choose an ordered basis {v₁, …, vₙ} for V. Each linear map ρ(g): V → V is then encoded as an n×n matrix whose columns are the coordinates of ρ(g)(v₁), …, ρ(g)(vₙ) in that basis. The homomorphism condition ρ(gh) = ρ(g)ρ(h) translates directly: the matrix product of the matrices for g and h equals the matrix for gh.

The immediate advantage is computability. To check whether a proposed assignment of matrices to group elements is a representation, you verify a finite number of matrix equations. For a group with generators g₁, …, gₖ and relations r₁, …, rₘ, you only need the matrices for the generators and must check that the relations hold at the matrix level. This reduces an infinite verification (all pairs g, h) to a finite one.

The cost is basis-dependence. The same abstract representation can look very different in two bases. For instance, a rotation of ℝ² by angle θ is diagonal in the basis of eigenvectors (with eigenvalues e^iθ and e^−iθ over ℂ) but has the familiar rotation matrix [[cos θ, −sin θ], [sin θ, cos θ]] in the standard basis. The representation is the same; only the coordinates changed. Two matrix representations related by ρ'(g) = Pρ(g)P⁻¹ for a fixed invertible P and all g ∈ G are called equivalent — they are different coordinate descriptions of the same abstract representation.

This means that properties worth studying are those invariant under conjugation. The determinant det(ρ(g)), the trace tr(ρ(g)), and the eigenvalues of ρ(g) are all basis-independent. The trace, in particular, will become the central object of study when you reach character theory: it distills each representation into a function on the group that captures essentially all the structural information, without any reference to a basis.