Linear Algebra II (pdf, 500 kB)

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5.1. Lemma. If p(x) is a polynomial (over F ) and λ ∈ F such that p(λ) = 0,
then (x − λ) m and p(x) are coprime for all m ≥ 1.
Proof. First, consider m = 1. Let
q(x) = p(x)
− 1 ;
p(λ)
this is a polynomial such that q(λ) = 0. Therefore, we can write q(x) = (x−λ)r(x)
with some polynomial r(x). This gives us
−r(x)(x − λ) + 1
p(x) = 1 .
p(λ)
Now, taking the mth power on both sides, we obtain an equation
−r(x) m(x − λ) m + a(x)p(x) = 1 .
Now we can feed this into Prop. 4.7.
5.2. Theorem. Let V be a finite-dimensional vector space, and let f : V → V be
an endomorphism whose characteristic polynomial splits into linear factors:
Pf(x) = (x − λ1) m1 · · · (x − λk) mk ,
where the λi are distinct. Then V = U1 ⊕ · · · ⊕ Uk, where Uj = ker(f − λ idV ) mj
is the generalized λj-eigenspace of f.
Proof. Write Pf(x) = p1(x) · · · pk(x) with pj(x) = (x − λj) mj . By Lemma 5.1, we
know that the pj(x) are coprime in pairs. By the Cayley-Hamilton Theorem 2.1,
we know that Pf(f) = 0. The result then follows from Prop. 4.7.
5.3. Theorem (Jordan Normal Form). Let V be a finite-dimensional vector
space, and let f : V → V be an endomorphism whose characteristic polynomial
splits into linear factors:
Pf(x) = (x − λ1) m1 · · · (x − λk) mk ,
where the λi are distinct. Then there is a basis of V such that the matrix representing
f with respect to that basis is a block diagonal matrix with blocks of the
form
⎛
λ 1 0 · · · 0
⎞
0
⎜0
⎜
⎜0
B(λ, m) = ⎜ .
⎝0
λ
0
.
0
1
λ
.
0
· · ·
· · ·
...
· · ·
0
0
.
λ
0 ⎟
0 ⎟
.
⎟ ∈ Mat(F, m)
⎟
1⎠
0 0 0 · · · 0 λ
where λ ∈ {λ1, . . . , λk}.
Proof. We keep the notations of Thm. 5.2. We know that on Uj, (f −λj id) mj = 0,
so f|Uj = λj idUj + gj, where g mj
j = 0, i.e., gj is nilpotent. By Thm. 3.3, there
is a basis of Uj such that gj is represented by a block diagonal matrix Bj with
blocks of the form B(0, m) (such that the sum of the m’s is mj). Therefore, f|Uj is
represented by Bj +λjIdim Uj , which is a block diagonal matrix composed of blocks
B(λj, m) (with the same m’s as before). The basis of V that is given by the union
of the various bases of the Uj then does what we want, compare Remark 4.4.