Linear Algebra, 2020a

402 Chapter Five. Similarity
II
Similarity
We’ve defined two matrices H and Ĥ to be matrix equivalent if there are
nonsingular P and Q such that Ĥ = PHQ. We were motivated by this diagram
showing H and Ĥ both representing a map h, but with respect to different pairs
of bases, B, D and ˆB, ˆD.
V wrt B
id
⏐
↓
V wrt ˆB
h
−−−−→
H
h
−−−−→
Ĥ
W wrt D
id
⏐
↓
W wrt ˆD
We now consider the special case of transformations, where the codomain
equals the domain, and we add the requirement that the codomain’s basis equals
the domain’s basis. So, we are considering representations with respect to B, B
and D, D.
t
V wrt B −−−−→ V wrt B
T
⏐
⏐
id↓
id↓
V wrt D
t
−−−−→
ˆT
V wrt D
In matrix terms, Rep D,D (t) =Rep B,D (id) Rep B,B (t) ( Rep B,D (id) ) −1
.
II.1
Definition and Examples
1.1 Example Consider the derivative transformation d/dx: P 2 → P 2 , and two
bases for that space B = 〈x 2 ,x,1〉 and D = 〈1, 1 + x, 1 + x 2 〉 We will compute
the four sides of the arrow square.
P 2 wrt B
id
⏐
↓
d/dx
−−−−→
T
P 2 wrt B
id
⏐
↓
P 2 wrt D
d/dx
−−−−→
ˆT
P 2 wrt D
The top is first. The effect of the transformation on the starting basis B
x 2 d/dx
↦−→ 2x
x d/dx
↦−→ 1
1 d/dx
↦−→ 0