Two matrix representations of the same group are called equivalent if they are related by:
AA permutation of the rows and columns of each matrix
BConjugation by a fixed invertible matrix P: ρ'(g) = Pρ(g)P⁻¹ for all g ∈ G
CMultiplying each matrix by a fixed scalar
DTransposing each matrix
Equivalent matrix representations differ by a change of basis. If we change from basis B to basis B' via an invertible matrix P, then the matrix of ρ(g) in the new basis is Pρ(g)P⁻¹. The same P is used for all group elements — this is what makes it a uniform change of basis rather than an arbitrary reshuffling. This corresponds to the abstract representations being related by an intertwining isomorphism.
Question 2 True / False
A matrix representation assigns a matrix to each group element such that the product of the matrices for g and h equals the matrix for gh.
TTrue
FFalse
Answer: True
This is the homomorphism condition written in matrix language: if ρ: G → GL_n(F) is a matrix representation, then ρ(g)ρ(h) = ρ(gh) for all g, h ∈ G. Matrix multiplication corresponds to composition of linear transformations, so this says the representation preserves the group operation. In particular, ρ(e) = Iₙ (the identity matrix) and ρ(g⁻¹) = ρ(g)⁻¹.
Question 3 Short Answer
Consider the representation of ℤ/2ℤ = {0, 1} on ℝ² given by ρ(0) = I₂ and ρ(1) = [[−1, 0], [0, 1]]. What does this representation do geometrically?
Think about your answer, then reveal below.
Model answer: It reflects vectors across the y-axis. The generator 1 maps to the matrix that negates the x-coordinate and preserves the y-coordinate, which is reflection through the line x = 0.
This illustrates how matrix representations encode geometric transformations. The group ℤ/2ℤ has order 2, so the non-identity element must square to the identity — and indeed the reflection matrix squares to I₂. Finding such concrete geometric interpretations is one of the primary benefits of working with matrix representations rather than abstract homomorphisms.
Question 4 True / False
If we change the basis of a 3-dimensional representation, the traces of the representing matrices change.
TTrue
FFalse
Answer: False
The trace of a matrix is invariant under conjugation: tr(PAP⁻¹) = tr(A). Since a change of basis replaces each matrix ρ(g) with Pρ(g)P⁻¹ for a fixed invertible P, the trace tr(ρ(g)) is unchanged. This basis-independence of the trace is precisely why it becomes the foundation of character theory — it extracts representation information that does not depend on coordinate choices.