Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 207
and finding a suitable string basis is routine.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 2 3 −1
B = 〈 ⎜0
⎟
⎝0⎠ , ⎜−1
⎟
⎝−1⎠ , ⎜ 3
⎟
⎝−6⎠ , ⎜−1
⎟
⎝ 1 ⎠ 〉
1 2 3 0
Five.IV.2.22 There are two eigenvalues, λ 1 = −2 and λ 2 = 1. The restriction of t + 2 to N ∞ (t + 2)
could have either of these actions on an associated string basis.
⃗β 1 ↦→ β ⃗ 2 ↦→ ⃗0 β1 ⃗ ↦→ ⃗0
⃗β 2 ↦→ ⃗0
The restriction of t − 1 to N ∞ (t − 1) could have either of these actions on an associated string basis.
⃗β 3 ↦→ β ⃗ 4 ↦→ ⃗0 β3 ⃗ ↦→ ⃗0
⃗β 4 ↦→ ⃗0
In combination, that makes four possible Jordan forms, the two first actions, the second and first, the
first and second, and the two second actions.
⎛
⎞ ⎛
⎞ ⎛
⎞ ⎛
⎞
−2 0 0 0 −2 0 0 0 −2 0 0 0 −2 0 0 0
⎜ 1 −2 0 0
⎟ ⎜ 0 −2 0 0
⎟ ⎜ 1 −2 0 0
⎟ ⎜ 0 −2 0 0
⎟
⎝ 0 0 1 0⎠
⎝ 0 0 1 0⎠
⎝ 0 0 1 0⎠
⎝ 0 0 1 0⎠
0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
Five.IV.2.23 The restriction of t + 2 to N ∞ (t + 2) can have only the action β ⃗ 1 ↦→ ⃗0. The restriction
of t − 1 to N ∞ (t − 1) could have any of these three actions on an associated string basis.
⃗β 2 ↦→ β ⃗ 3 ↦→ β ⃗ 4 ↦→ ⃗0 β2 ⃗ ↦→ β ⃗ 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗0
⃗β 2 ↦→ ⃗0
⃗β 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗0
Taken together there are three possible Jordan forms, the one arising from the first action by t − 1
(along with the only action from t + 2), the one arising from the second action, and the one arising
from the third action. ⎛
⎜
−2
⎜⎝
0 0 0
⎞ ⎛
⎞ ⎛
⎞
−2 0 0 0 −2 0 0 0
0 1 0 0
⎟
0 1 1 0⎠
0 0 1 1
⎜
⎝
0 1 0 0
⎟
0 1 1 0⎠
0 0 0 1
⎜
⎝
0 1 0 0
⎟
0 0 1 0⎠
0 0 0 1
Five.IV.2.24 The action of t + 1 on a string basis for N ∞ (t + 1) must be β ⃗ 1 ↦→ ⃗0. Because of the
power of x − 2 in the minimal polynomial, a string basis for t − 2 has length two and so the action of
t − 2 on N ∞ (t − 2) must be of this form.
⃗β 2 ↦→ β ⃗ 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗0
Therefore there is only one Jordan form that is possible.
⎛
⎞
−1 0 0 0
⎜ 0 2 0 0
⎟
⎝ 0 1 2 0⎠
0 0 0 2
Five.IV.2.25 There are two possible Jordan forms. The action of t + 1 on a string basis for N ∞ (t + 1)
must be β ⃗ 1 ↦→ ⃗0. There are two actions for t − 2 on a string basis for N ∞ (t − 2) that are possible with
this characteristic polynomial and minimal polynomial.
⃗β 2 ↦→ ⃗ β 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗ β 5 ↦→ ⃗0
The resulting Jordan form matrics are these.
⎛
⎞
−1 0 0 0 0
0 2 0 0 0
⎜ 0 1 2 0 0
⎟
⎝ 0 0 0 2 0⎠
0 0 0 1 2
⃗β 2 ↦→ ⃗ β 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗0
⃗β 5 ↦→ ⃗0
⎛
⎞
−1 0 0 0 0
0 2 0 0 0
⎜ 0 1 2 0 0
⎟
⎝ 0 0 0 2 0⎠
0 0 0 0 2