Linear Algebra, 2020a

430 Chapter Five. Similarity
Observe also that although n is not the zero map, the function n 2 = n ◦ n is
the zero map.
2.5 Example A linear function ˆn: C 4 → C 4 whose action on E 4 is given by the
string
⃗e 1 ↦→ ⃗e 2 ↦→ ⃗e 3 ↦→ ⃗e 4 ↦→ ⃗0
has R(ˆn) ∩ N (ˆn) equal to the span [{⃗e 4 }], has R(ˆn 2 ) ∩ N (ˆn 2 )=[{⃗e 3 ,⃗e 4 }], and
has R(ˆn 3 ) ∩ N (ˆn 3 )=[{⃗e 4 }]. The matrix representation is all zeros except for
some subdiagonal ones.
⎛
⎞
0 0 0 0
1 0 0 0
ˆN = Rep E4 ,E 4
(ˆn) = ⎜
⎟
⎝0 1 0 0⎠
0 0 1 0
Although ˆn is not the zero map, and neither is ˆn 2 or ˆn 3 , the function ˆn 4 is the
zero function.
2.6 Example Transformations can act via more than one string. The transformation
t acting on a basis B = 〈⃗β 1 ,...,⃗β 5 〉 by
⃗β 1 ↦→ ⃗β 2 ↦→ ⃗β 3 ↦→ ⃗0
⃗β 4 ↦→ ⃗β 5 ↦→ ⃗0
will have, for instance, ⃗β 3 in the intersection of its range space and null space.
The strings make clear that t 3 is the zero map. This map is represented by a
matrix that is all zeros except for blocks of subdiagonal ones
⎛
⎞
0 0 0 0 0
1 0 0 0 0
Rep B,B (t) =
0 1 0 0 0
⎜
⎟
⎝0 0 0 0 0⎠
0 0 0 1 0
(the lines just visually organize the blocks).
In those examples all vectors are eventually transformed to zero.
2.7 Definition A nilpotent transformation is one with a power that is the zero
map. A nilpotent matrix is one with a power that is the zero matrix. In either
case, the least such power is the index of nilpotency.
2.8 Example In Example 2.4 the index of nilpotency is two. In Example 2.5 it is
four. In Example 2.6 it is three.