Linear Algebra, 2020a

Section IV. Jordan Form 457
2.12 Example This matrix
⎛ ⎞
2 0 1
⎜ ⎟
T = ⎝0 6 2⎠
0 0 2
has the characteristic polynomial (x − 2) 2 (x − 6).
2 − x 0 1
|T − xI| =
0 6− x 2
=(x − 2) 2 (x − 6)
∣ 0 0 2− x∣
First we do the eigenvalue 2. Computation of the powers of T − 2I, and of
the null spaces and nullities, is routine. (Recall our convention of taking T to
represent a transformation t: C 3 → C 3 with respect to the standard basis.)
p (T − 2I) p N ((t − 2) p ) nullity
⎛ ⎞ ⎛ ⎞
0 0 1
x
1 ⎜
⎝0 4 2⎟
{ ⎜
⎠ ⎝0⎟
⎠ | x ∈ C} 1
0 0 0
0
⎛ ⎞ ⎛ ⎞
0 0 0 x
2 ⎜
⎝0 16 8⎟
⎠
{ ⎜
⎝−z/2⎟
| x, z ∈ C} 2
⎠
0 0 0 z
⎛ ⎞
0 0 0
3 ⎜
⎝0 64 32⎟
–same– –same–
⎠
0 0 0
So the generalized null space N ∞ (t − 2) has dimension two. We know that the
restriction of t − 2 is nilpotent on this subspace. From the way that the nullities
grow we know that the action of t − 2 on a string basis is ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗0. Thus
we can represent the restriction in the canonical form
⎛ ⎞ ⎛
( )
1
0 0
⎜ ⎟ ⎜
−2
⎞
⎟
N 2 = = Rep
1 0
B,B (t − 2) B 2 = 〈 ⎝ 1 ⎠ , ⎝ 0 ⎠〉
−2 0
(other choices of basis are possible). Consequently, the action of the restriction
of t to N ∞ (t − 2) is represented by this Jordan block.
( )
2 0
J 2 = N 2 + 2I = Rep B2 ,B 2
(t) =
1 2