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generalized eigenvector∗
CWoo†
2013-03-21 23:29:38
Let V be a vector space over a field k and T a linear transformation on V (a
linear operator). A non-zero vector v ∈ V is said to be a generalized eigenvector
of T (corresponding to λ) if there is a λ ∈ k and a positive integer m such that
(T − λI)m (v) = 0,
where I is the identity operator.
In the equation above, it is easy to see that λ is an eigenvalue of T . Suppose
that m is the least such integer satisfying the above equation. If m = 1, then
λ is an eigenvalue of T . If m > 1, let w = (T − λI)m−1 (v). Then w 6= 0 (since
v 6= 0) and (T − λI)(w) = 0, so λ is again an eigenvalue of T .
Let v be a generalized eigenvector of T corresponding to the eigenvalue λ.
We can form a sequence
v, (T − λI)(v), (T − λI)2 (v), . . . , (T − λI)i (v), . . . , (T − λI)m (v) = 0, 0, . . .
The set Cλ (v) of all non-zero terms in the sequence is called a cycle of generalized
eigenvectors of T corresponding to λ. The cardinality m of Cλ (v) is its length.
For any Cλ (v), write vλ = (T − λI)m−1 (v).
Below are some properties of Cλ (v):
• vλ is the only eigenvector of λ in Cλ (v), for otherwise vλ = 0.
• Cλ (v) is linearly independent.
Pm
Proof. Let vi = (T − λI)i−1 (v), where i = 1, . . . , m. Let 0 = i=1 ri vi
with ri ∈ k. Induct on i. If i = 1, then v1 = v 6= 0, so r1 = 0 and {v1 }
is linearly independent. Suppose the property is P
true when i = m − 1.
m
Apply T − λI to the equation, and we have 0 = i=1 ri (T − λI)(vi ) =
Pm−1
i=1 ri vi+1 . Then r1 = · · · = rm−1 = 0 by induction. So 0 = rm vm =
rm vλ and thus rm = 0 since vλ is an eigenvector and is non-zero.
∗ hGeneralizedEigenvectori created: h2013-03-21i by: hCWooi version: h39754i Privacy
setting: h1i hDefinitioni h65F15i h65-00i h15A18i h15-00i
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• More generally, it can be shown that Cλ (v1 ) ∪ · · · ∪ Cλ (vk ) is linearly
independent whenever {v1λ , . . . , vkλ } is.
• Let E = span(Cλ (v)). Then E is a (m + 1)-dimensional subspace of the
generalized eigenspace of T corresponding to λ. Furthermore, let T |E be
the restriction of T to E, then [T |E ]Cλ (v) is a Jordan block, when Cλ (v)
is ordered (as an ordered basis) by setting
(T − λI)i (v) < (T − λI)j (v)
whenever
i > j.
Indeed, for if we let wi = (T − λI)m+1−i (v) for i = 1, . . . m + 1, then
λwi
if i = 1,
T (wi ) = (T − λI + λI)(T − λI)m+1−i (v) =
wi−1 + λwi otherwise.
so that [T |E ]Cλ (v) is the (m + 1) × (m + 1) matrix given by

λ
0

0

 ..
.

0
0
1
λ
0
..
.
0
1
λ
..
.
···
···
···
..
.
0
0
0
0
···
···

0
0

0

.. 
.

1
λ
• A cycle of generalized eigenvectors is called maximal if v ∈
/ (T − λI)(V ).
If V is finite dimensional, any cycle of generalized eigenvectors Cλ (v) can
always be extended to a maximal cycle of generalized eigenvectors Cλ (w),
meaning that Cλ (v) ⊆ Cλ (w).
• In particular, any eigenvector v of T can be extended to a maximal cycle of generalized eigenvectors. Any two maximal cycles of generalized
eigenvectors extending v span the same subspace of V .
References
[1] Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
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