Linear Algebra, 2020a

438 Chapter Five. Similarity
Then add ⃗β 2 , ⃗β 4 ∈ N (n 2 ) such that n(⃗β 2 )=⃗β 3 and n(⃗β 4 )=⃗β 5 .
⎛ ⎞
0
1
⃗β 2 =
0
⎜ ⎟
⎝0⎠
0
Finish by adding ⃗β 1 such that n(⃗β 1 )=⃗β 2 .
Exercises
⎛ ⎞
1
0
⃗β 1 =
1
⎜ ⎟
⎝0⎠
0
⎛ ⎞
0
1
⃗β 4 =
0
⎜ ⎟
⎝1⎠
0
̌ 2.20 What is the index of nilpotency of the right-shift operator, here acting on the
space of triples of reals?
(x, y, z) ↦→ (0, x, y)
̌ 2.21 For each string basis state the index of nilpotency and give the dimension of
the range space and null space of each iteration of the nilpotent map.
(a) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗0
⃗β 3 ↦→ ⃗β 4 ↦→ ⃗0
(b) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗β 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗0
⃗β 5 ↦→ ⃗0
⃗β 6 ↦→ ⃗0
(c) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗β 3 ↦→ ⃗0
Also give the canonical form of the matrix.
2.22 Decide which of these matrices are nilpotent.
⎛ ⎞
( ) ( ) −3 2 1
−2 4 3 1
(a)
(b)
(c) ⎝−3 2 1⎠
−1 2 1 3
−3 2 1
⎛
⎞
45 −22 −19
(e) ⎝33 −16 −14⎠
69 −34 −29
̌ 2.23 Find the canonical form of this matrix.
⎛
⎞
0 1 1 0 1
0 0 1 1 1
0 0 0 0 0
⎜
⎟
⎝0 0 0 0 0⎠
0 0 0 0 0
(d)
⎛
1 1
⎞
4
⎝3 0 −1⎠
5 2 7