2.4 Linear transformation
with respect to a B-basis
xA = PDP−1⟶yP−1 ↓ ↑ PP−1 ↓ ↑ P[x]B[T]B = D⟶[y]B
x = P[x]B
P−1x = [x]B
P−1x = [x]B
Let’s start from y:
y = P[y]B
= PD[x]B
= PDP−1x
= PD[x]B
= PDP−1x
Therefore
P−1AP = D
Theorem 8
Diagonal matrix representation
Suppose A = PDP−1, where D is a diagonal n × n matrix. If B is the basis for ℝn formed from the columns of P, then D is the B-matrix for the transformation x ↦ Ax.
Proof
Denote the columns of P by b₁, …, bₙ, so that B = {b₁, …, bₙ}. In this case P is the change-of-coordinates matrix.
[T]B = [[T(b₁)]B ⋯ [T(bₙ)]B]
= [P−1Ab₁ ⋯ P−1Abₙ]
= P−1A[b₁ ⋯ bₙ]
= P−1AP = D ∎
= [P−1Ab₁ ⋯ P−1Abₙ]
= P−1A[b₁ ⋯ bₙ]
= P−1AP = D ∎
