Linear Algebra, 2020a

406 Chapter Five. Similarity
1.6 Example 1.4 shows that the only matrix similar to a zero matrix is itself and
that the only matrix similar to the identity is itself.
(a) Show that the 1×1 matrix whose single entry is 2 is also similar only to itself.
(b) Is a matrix of the form cI for some scalar c similar only to itself?
(c) Is a diagonal matrix similar only to itself?
̌ 1.7 Consider this transformation of C
⎛ ⎞
3
⎛ ⎞
x x − z
t( ⎝ ⎠) = ⎝
and these bases.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 0
1 1 1
B = 〈 ⎝2⎠ , ⎝1⎠ , ⎝0⎠〉 D = 〈 ⎝0⎠ , ⎝1⎠ , ⎝0⎠〉
3 0 1
0 0 1
y
z
We will compute the parts of the arrow diagram to represent the transformation
using two similar matrices.
(a) Draw the arrow diagram, specialized for this case.
(b) Compute T = Rep B,B (t).
(c) Compute ˆT = Rep D,D (t).
(d) Compute the matrices for the other two sides of the arrow square.
1.8 Consider the transformation t: P 2 → P 2 described by x 2 ↦→ x + 1, x ↦→ x 2 − 1,
and 1 ↦→ 3.
(a) Find T = Rep B,B (t) where B = 〈x 2 ,x,1〉.
(b) Find ˆT = Rep D,D (t) where D = 〈1, 1 + x, 1 + x + x 2 〉.
(c) Find the matrix P such that ˆT = PTP −1 .
z
2y
̌ 1.9 Let T represent t: C 2 → C 2 with respect to B, B.
( ) ( ( 1 −1
1 1
T =
B = 〈 , 〉, D = 〈
2 1
0)
1)
⎠
( 2
0)
( ) 0
, 〉
−2
We will convert to the matrix representing t with respect to D, D.
(a) Draw the arrow diagram.
(b) Give the matrix that represents the left and right sides of that diagram, in
the direction that we traverse the diagram to make the conversion.
(c) Find Rep D,D (t).
̌ 1.10 Exhibit a nontrivial similarity relationship by letting t: C 2 → C 2 act in this
way,
( ( ) ( )
1 3 −1
↦→
↦→
2)
0 1
( ) −1
2
picking two bases B, D, and representing t with respect to them, ˆT = Rep B,B (t)
and T = Rep D,D (t). Then compute the P and P −1 to change bases from B to D
and back again.
̌ 1.11 Show that these matrices are not similar.
⎛ ⎞ ⎛ ⎞
1 0 4 1 0 1
⎝
1 1 3
2 1 7
1.12 Explain Example 1.4 in terms of maps.
⎠
⎝
0 1 1
3 1 2
⎠