Page 262 - Matrix Analysis & Applied Linear Algebra

4.8 Change of Basis and Similarity 257
Exercises for section 4.8

4.8.1. Explain why rank is a similarity invariant.

4.8.2. Explain why similarity is transitive in the sense that A - B and B - C
implies A - C.

3
4.8.3. A(x, y, z)=(x +2y − z, −y, x +7z) is a linear operator on .
(a) Determine [A] S , where S is the standard basis.
(b) Determine [A] S as well as the nonsingular matrix Q such that
1 1 1

[A] S = Q −1 [A] S Q for S = 0 , 1 , 1 .

0 0 1
1 2 0 1 1 1

4.8.4. Let A = 3 1 4 and B = 1 , 2 , 2 . Consider A
0 1 5 1 2 3
n×1
as a linear operator on by means of matrix multiplication A(x)=
Ax, and determine [A] B .

4 6 −2 −3
4.8.5. Show that C = and B = are similar matrices, and
3 4 6 10
find a nonsingular matrix Q such that C = Q −1 BQ. Hint: Consider
2
B as a linear operator on , and compute [B] S and [B] S , where S
( 2 −3 )
is the standard basis, and S = , .

−1 2
4.8.6. Let T be the linear operator T(x, y)=(−7x − 15y, 6x +12y). Find
2 0
a basis B such that [T] B = , and determine a matrix Q such
0 3
that [T] B = Q −1 [T] S Q, where S is the standard basis.

4.8.7. By considering the rotator P(x, y)=(x cos θ − y sin θ, x sin θ + y cos θ)
described in Example 4.7.1 and Figure 4.7.1, show that the matrices

iθ
cos θ − sin θ e 0
R = and D = −iθ
sin θ cos θ 0 e
are similar over the complex field. Hint: In case you have forgotten (or
didn’t know), e iθ = cos θ + i sin θ.