Linear Algebra, 2020a

404 Chapter Five. Similarity
That leaves the bottom of the square. There are two ways to compute the
matrix ˆT. One is to compute it directly by finding the effect of the transformation
on elements of D
1 d/dx
↦−→ 0
1+ x d/dx
↦−→ 1
1+ x 2 d/dx
↦−→ 2x
represented with respect to D.
⎛
⎜
0 1 −2
⎞
⎟
ˆT = Rep D,D (d/dx) = ⎝0 0 2 ⎠
0 0 0
The other way to compute ˆT, and this is the way we will usually do it, is to
follow the diagram up, over, and then down.
Rep D,D (d/dx) =Rep B,D (id) Rep B,B (d/dx) Rep D,B (id)
ˆT = Rep B,D (id) T Rep D,B (id)
⎛
⎞ ⎛
−1 −1 1
⎜
⎟ ⎜
0 0 0
⎞ ⎛
⎟ ⎜
0 0 1
⎞
⎟
= ⎝ 0 1 0⎠
⎝2 0 0⎠
⎝0 1 0⎠
1 0 0 0 1 0 1 1 1
Multiplying out gives the same matrix ˆT as we found above.
1.2 Definition The matrices T and ˆT are similar if there is a nonsingular P such
that ˆT = PTP −1 .
Since nonsingular matrices are square, T and ˆT must be square and of the same
size. Exercise 15 checks that similarity is an equivalence relation.
1.3 Example The definition does not require that we consider a map. Calculation
with these two
(
P =
2 1
1 1
)
(
T =
gives that T is similar to this matrix.
( )
12 −19
ˆT =
7 −11
2 −3
1 −1
1.4 Example The only matrix similar to the zero matrix is itself: PZP −1 = PZ = Z.
The identity matrix has the same property: PIP −1 = PP −1 = I.
)