Lineáris transzformációk invariáns alterei

2

V F A V V
M A
v ∈ M A(v) M A(M) ⊆ M.
M
V
w A(w) M w M
M M
A A
A
A
M A V A
M
V = {x1, . . . , xr, xr+1, . . . , xn} V {x1, . . . , xr}
M A
AV =
A11 A12
0 A22
A11 A M r × r {x1, . . . , xr}
0 n − r × r A12, A22 r × n − r
n − r × n − r N = (xr+1, . . . , xn) A
A12
A11
AV =
0
0
A22
A11 A22
M A
A V A M
N (M, N)
A
A
ker(A) (A) V
A
v ∈ V {v, A(v), A2 (v), . . .}
(v, A) V n
{v, A(v), A2 (v), . . . An (v)}
n + 1
Ak (v) (1 ≤ k ≤ n) v, A(v), . . . , Ak−1 (v)
m Ak+m (v)
v, A(v), . . . , Ak−1 (v) m
A k k−1
(v) = αiA i (v) ,
i=0

A k+1 k−1
(v) = αiA i+1 k−2
(v) = αiA i+1 (v) + αk−1
i=0
i=0
k−1
αiA i
(v) =
i=0
k−1
= αk−1α0v + (αi−1 + αk−1αi)A i (v) ,
i=1
m = 1 m − 1 ≥ 1
A k+m−1 k−1
(v) = βiA i (v) ,
i=0
A k+m k−1
(v) = βiA i+1 k−2
(v) = βiA i+1 (v) + βk−1
i=0
i=0
k−1
k−1
= βk−1β0v + (βi−1 + βk−1αi)A i (v) ,
i=1
αiA i
(v) =
A k+m (v) v, A(v), . . . , A k−1 (v)
(v, A) = v, A(v), . . . , A k−1 (v)
(v, A)
(v, A) A v
A ∈ L(V )
p(A) = α0A 0 + α1A + · · · + αkA k
V A
p(A)
p(A)
A v ∈ ker(p(A)) p(A)(v) = 0
p(A)(A(v)) = A(p(A)(v)) = A(0) = 0 A(v) ∈ ker(p(A))
v ∈ (p(A)) w ∈ V p(A)(w) = v
p(A)(A(w)) = A(p(A)(w)) = A(v) A(v) ∈ (p(A))
p(t) ∈ F[t] F → F
i=0

F F V
L(V ) F dim V × dim V
F dim V ×dim V L(V )
dim V ×dim V
Φ : L(V ) → F
A ∈ L(V )
p(t) = α0 + α1t + · · · + αnt n
V
p(A) = α0A 0 + α1A + · · · + αnA n
A ∈ F dim V ×dim V
p(A) = α0A 0 + α1A + · · · + αnA n dim V ×dim V
∈ F
p(A) A
p(A) A
A
p(t)
A A p(A)
p(t) + q(t) = r(t) p(t) · q(t) = s(t) ,
p(A) + q(A) = r(A) p(A) · q(A) = s(A) ,
p(A) + q(A) = r(A) p(A) · q(A) = s(A).
A A p(t)
p(A) = 0 p(A) = 0).
p(t)
B p(t)
p(B −1 AB) = B −1 p(A)B .

V n F A ∈ L(V )
F[t] dim L(V ) = n 2
A 0 = I, A, . . . , A n2
n 2 +1
α0A 0 + α1A + · · · + αn2A n2
= 0 .
p(t) = α0 + α1t + · · · + αn2t n2
A A
p(t) = α0 + α1t + · · · + αmtm m 1/αm · p(t)
m
A
A
mA(t) m ′ A
A mA(t) − m ′ A (t) mA(t) − m ′ A
(t) A
mA(t) mA(t)
(t)
A
A
f(t) A
f(A) = 0
f(t) = h(t)mA(t) + r(t) 0 ≤ deg r(t) < deg mA(t).
A
0 = f(A) = h(A)mA(A) + r(A) = r(A) ,
r(t) ≡ 0 , A mA(t)
✷

V n F
A ∈ L(V ) mA(t) ∈ F[t]
k V k A
p(t) k A
ker(p(A)) k A k A
mA(t) A mA(t) =
p(t)q(t) p(t) k
ker(p(A)) A v = 0
ker(p(A)) q(t) A
(v, A) k A ker(p(A))
V {v, A(v), . . . A n (v)}
m(≤ n)
{v, A(v), . . . , A m−1 (v)}
A m (v) v, A(v), . . . , A m−1 (v)
A m (v) = ε0v + ε1A(v) + · · · + εm−1A m−1 (v)
s(t) = −ε0 − ε1t − · · · − εm−1t m−1 + t m
s(A)(v) = 0 , m
m A
v v ker(p(A))
p(A)(v) = 0 ,
m = deg s(t) ≤ deg p(t) = k .
p(t) s(t)
p(t) = h(t)s(t) + r(t) 0 ≤ deg r(t) < deg s(t) = m ,
p(A) = h(A)s(A) + r(A) .
0 = p(A)(v) = h(A)s(A)(v) + r(A)(v) = r(A)(v) ,
r(t) ≡ 0 p(t) = h(t)s(t) .
p(t) h(t) s(t) p(t)
k = m . (v, A) = v, A(v), . . . , A k−1 (v)
k A
ker(p(A)) v w
(v, A) (w, A)
x ∈ (v, A) (w, A)

{x, A(x), . . . , A k−1 (x)}
(v, A) (w, A)
A
dim(v, A) = dim(w, A) = k
{x, A(x), . . . , A k−1 (x)} (v, A)
(w, A)
(v, A) = (w, A) .
ker(p(A))
v1 ∈ ker(p(A)) (v1, A)
ker(p(A)) \ (v1, A) v2
(v1, A) ⊕ (v2, A) 2 · k ker(p(A))
v3 ∈ ker(p(A)) \ (v1, A) ⊕ (v2, A)
(v1, A) ⊕ (v2, A) ⊕ (v3, A)
ker(p(A))
r ≥ 1
ker(p(A)) \ (v1, A) ⊕ · · · ⊕ (vr, A) = ∅ ,
x ∈ ker(p(A)) x = 0 (y, A) ⊆
ker(p(A)) (x, A) = (y, A)
vi ∈ (v1, A) ⊕ · · · ⊕ (vi−1, A) ,
(vi, A) (v1, A) ⊕ · · · ⊕ (vi−1, A) = {0} .
x
x ∈ (vi, A) (v1, A) ⊕ · · · ⊕ (vi−1, A),
(vi, A) = (x, A) x = i−1 j=1 wj,
(vj, A) j = (1, . . . , i − 1).
A
wj ∈
t i−1
(x) = A t (wj) ∈ (v1, A) ⊕ · · · ⊕ (vi−1, A)
j=1
t(= 1, . . . , k − 1)
vi ∈ (x, A) ⊆ (v1, A) ⊕ · · · ⊕ (vi−1, A),

vi ∈ (v1, A)⊕· · ·⊕(vi−1, A)
✷
2
1
1
F V A ∈ L(V )
A mA(t) λ ∈ F t − λ
mA(t) ker(A−λI) 1
A 1 A
A F
A λ A ker(A − λI)
s A
ker(A − λI) λ
ker(A − λI) λ
A F V
F F
A ∈ L(V ) (V = {0})
s
A(s) = (A − λI)(s) + λs = λs
A−λI V
v ker(A − λI) mA(t) t − λ ∈ F[t]
mA(t) = q(t)(t − λ) + γ
0
mA(A) = q(A)(A − λI) + γI

0 = mA(A)(v) = q(A)(A − λI)(v) + γI(v) = γv ,
γ = 0 t−λ mA(t) mA(λ) = 0 ,
λ
V F A A
λ ∈ F λ
A A − λI ∈ L(V )
0
A ∈ L(V ) λ
M A V M ◦ A ′
V ′
A ∈ L(V ) λ
p(t) ∈ F[t] p(A) p(λ)
A A −1 1
λ .
(A − λI) ′
= A ′
− λI ′
,
A − λI A ′
− λI ′
v M y M ◦
[v, A ′ (y)] = [A(v), y],
A(v) ∈ M, A ′ (y) ∈ M ◦ A ′ (y) ∈ M ◦ , A(v) ∈ M
A λ
s(= 0) ∈ V A(s) = λs . A2 (s) =
A(A(s)) = A(λs) = λA(s) = λ2s ,
k Ak (s) = λks
p(t) = α0 + α1t + · · · + αktk ∈ F[t]
p(A)(s) = (α0I + α1A + · · · + αkA k )(s) = α0s + α1A(s) + · · · + αkA k (s) =

α0s + α1λs + · · · + αkλ k s = p(λ)s,
p(λ) p(A)
s = A −1 (A(s)) = A −1 (λs) = λA −1 (s)
λ = 0
A −1 (s) = 1
s .
λ
3 V A
V = {v1, v2, v3}
A(v1) = v2 + v3, A(v2) = v1 + v3, A(v3) = v1 + v2
A A − λI V
⎡
A = ⎣
0 1 1
1 0 1
1 1 0
⎤
⎡
⎦ A − λE = ⎣
−λ 1 1
1 −λ 1
1 1 −λ
λ A − λE
λ
A − λE
ξ1, ξ2, ξ3
ξ1 ξ2 ξ3
v1 −λ 1 1
v2 −λ 1
v3 1 1 −λ
−→
λ = −1
⎡
ξ1
⎢
⎣ ξ2
⎤ ⎡
−1
⎥ ⎢
⎦ = ⎣ 1
⎤
−1
⎥
0 ⎦ ·
0 1
ξ3
ξ2
ξ3
v1 1 − λ 2 1 + λ
ξ1 −λ 1
v3 1 + λ −λ − 1
τ2
τ2, τ3
s1 =
⎡
⎢
⎣
−1
1
0
⎤
⎡
τ3
⎥
⎢
⎦ s2 = ⎣
−1
0
1
⎤
⎥
⎦
⎤
⎦

λ = −1
λ = −1
ξ3
v1 2 + λ − λ 2
ξ1
ξ2
1 − λ
λ = 2
⎡ ⎤ ⎡ ⎤
ξ1 1
⎢ ⎥ ⎢ ⎥
⎣ ξ2 ⎦ = ⎣ 1 ⎦ · τ3 (τ3 = 0) .
ξ3 1
λ = 2
⎡ ⎤
1
⎢ ⎥
s3 = ⎣ 1 ⎦ .
1
A
S = {s1 = −v1 + v2, s2 = −v1 + v3, s3 = v1 + v2 + v3}
A(s1) = −s1, A(s2) = −s2 A(s3) = 2 · s3 ,
⎡
⎤
−1 0 0
AS = ⎣ 0 −1 0 ⎦
0 0 2
A
n F V A ∈ L(V )
S = {s1, . . . , sn} A
S
λ1, . . . , λn .
−1
A(si) = λisi (i = 1, . . . , n) ,

⎡
⎢
A(si)S = ⎢
⎣
λi
⎥
0 ⎥
⎥
⎦
0
i λi A
AS i A(si)S i(= 1, . . . , n)
AS =
⎡
⎢ ⎢ ⎣
0
0
⎤
⎥ ,
λ1 0 . . . 0
0 λ2 . . . 0
0 0 . . . λn
−1
V A λ1, . . . , λk
s1, . . . , sk
k = 1
λ1, . . . , λk−1
{s1, . . . , sk−1}
λk sk
k−1
(a) sk =
i=1
αisi ,
{s1, . . . , sk−1, sk}
A sk
⎤
⎥
⎦

k−1
k−1
(b) λksk = A(sk) = αiA(si) =
i=1
i=1
αiλisi .
λk
k−1
0 = αi(λi − λk)si
i=1
sk = 0 αi
λi −
λk, (i = 1, . . . , k − 1)
{s1, . . . , sk−1}
{s1, . . . , sk−1, sk}
n V A n
A
A
A
A
A
V F A ∈ L(V )
A mA(t) p(t)
q(t)
V = ker p(A) ⊕ ker q(A) ,
V A
p(t) ker p(A) A
V ker p(A) ker q(A)
V
ker p(A) ker q(A)

u ∈ ker p(A) ∩ ker q(A). p(t) q(t)
1 1 = f(t)p(t)+g(t)q(t)
A
I(= 1 · A 0 ) = f(A)p(A) + g(A)q(A) .
u = I(u) = f(A)p(A)(u) + g(A)q(A)(u) = 0,
u ker p(A) ker q(A)
V ker p(A)
ker q(A)
I(= 1 · A 0 ) = f(A)p(A) + g(A)q(A)
v ∈ V
v = I(v) = (f(A)p(A))(v) + (g(A)q(A))(v)
q(A)(f(A)p(A)(v)) = f(A)(mA(A)(v)) = 0
p(A)(g(A)q(A)(v)) = g(A)(mA(A)(v)) = 0 , f(A)p(A)(v) ∈
ker q(A) g(A)q(A)(v) ∈ ker p(A) , v
✷
A V
ker p(A) ker q(A) V =
{v1, . . . , vr} ker p(A) W = {w1, . . . , ws} ker q(A)
A V {v1, . . . , vr, w1, . . . , ws}
A =
A1 0
0 A2
A1 r × r A2 s × s
A A A1 A2
A
A(vi) =
A(wk) =
r
αijvj (i = 1, . . . , r)
j=1
s
βkℓwℓ (ℓ = 1, . . . , s) ,
ker p(A) ker q(A) A
ℓ=1

mA(t) = p(t)q(t) A ∈ L(V )
p(t) A
ker p(A) q(t) A
ker q(A)
A ker p(A) p(t)
s(t) s(t)q(t)
A s(t)q(t)
mA(t)
v ∈ V
v = x + y (x ∈ ker p(A) , y ∈ ker q(A))
s(A)q(A)
v
(s(A)q(A)(v) = s(A)q(A)(x) + s(A)q(A)(y) = 0 ,
s(A)(x) = 0 q(A)(y) = 0
v ∈ V s(A)q(A) = 0 .
q(t) A
ker q(A) ✷
A ∈ L(V )
mA(t) = p1(t) · p2(t) · · · · · pr(t)
V
A
ker p1(A), ker p2(A), . . . , ker pr(A)
V = ker p1(A) ⊕ ker p2(A) ⊕ · · · ⊕ ker pr(A)
A ker pi(A)
⎡
A1 0 . . . 0
⎢ 0 A2 . . . 0
A = ⎢
⎣
0 0 . . . Ar
Ai A ker pi(A)
⎤
⎥
⎦

F[t] F
V A mA(t)
mA(t) = (p1(t)) m1 · · · · · (pr(t)) mr
V
V = ker p m1
1 (A) ⊕ · · · ⊕ ker pmr r (A)
A
ker p mi
i (A) = Vi (i = 1, . . . , r) ,
pi(A) Vi Bi Bi Vi
mi
B m
B m = 0 B
B m
pi(t) ki Vi
ki · mi .
A ∈ L(V ) p m (t) , p(t)
k V k · m .
A p m (t) , v ∈ V
p m−1 (A)(v) = 0 .
v, A(v), . . . , Ak−1 (v),
p(A)(v), (p(A)A)(v), . . . , (p(A)Ak−1 )(v),
p m−1 (A)(v), (p m−1 (A)A)(v), . . . , (p m−1 (A)A k−1 )(v)
m−1
i=0 j=0
k−1
βij(p i (A)A j )(v) = 0
ℓ(≥ 0)
j βℓj = 0 .
pm−ℓ−1 (A)
p m−ℓ−1 ⎛
m−1 k−1
(A) ⎝
i=0 j=0
βij(p i (A)A j )(v)
⎞
k−1
⎠ =
j=0
βℓj(p m−1 (A)A j )(v) = 0 ,

i < ℓ βij = 0, i > ℓ p m−ℓ+i−1 (A) = 0 j
βℓj = 0 . p m−1 (A)(v) = 0
p(A) p(t) k
p m−1 (A)(v), A p m−1 (A)(v) , . . . , A k−1 p m−1 (A)(v)
βij
V m · k
m · k .
n V A ∈ L(V )
n 2
V n A
n
A
mA(t) = (p1(t)) m1 · · · · · (pr(t)) mr
V
V = ker(p m1
1 (A)) ⊕ · · · ⊕ ker(pmr r (A))
deg pi(t) = ki (i = 1, . . . , r)
deg mA(t) = m1 · k1 + · · · + mr · kr ≤
≤ dim ker(p m1
1 (A)) + · · · + dim ker(pmr r (A)) = dim V = n .
n V n
3 V A
V = {v1, v2, v3} A(v1) = v1 +
v2 + v3, A(v2) = v1 + v2 A(v3) = v1 A

A
L(V ) 3 × 3
A V
⎡
A 0 = ⎣
⎡
A 2 = ⎣
1 0 0
0 1 0
0 0 1
3 2 1
2 2 1
1 1 1
⎤
⎡
⎦ A = ⎣
⎤
⎡
⎦ A 3 = ⎣
1 1 1
1 1 0
1 0 0
6 5 3
5 4 2
3 2 1
M3×3 = {Eij (i, j = 1, 2, 3)}
Eij 3 × 3 i j 1
A 0 A A 2 A 3
E11 1 1 3 6
E12 0 1 2 5
E13 0 1 1 3
E21 0 1 2 5
E22 1 1 2 4
E23 0 0 1 2
E31 0 1 1 3
E32 0 0 1 2
E33 0 1 1
A 2 A 3
E11 1 2
E12 1 2
E13 0 0
E21 1 2
E22 0 0
E23 2
A 1 3
E32 1 2
A 0 1 1
→
→
⎤
⎦
⎤
⎦
A A 2 A 3
E11 1 2 5
E12 1 2 5
E13 1 1 3
E21 1 2 5
E22 1 1 3
E23 0 1 2
E31 1 3
E32 0 1 2
A 0 0 1 1
A 3
E11 0
E12 0
E13 0
E21 0
E22 0
A2 2
A 1
E32 0
A0 −1
A 3 = 2A 2 +A−A 0 ,
A 3 − 2A 2 − A + A 0 = 0 , mA(t) =
t 3 − 2t 2 − t + 1 . ✷
→

V F A
v ∈ V pv(t) ∈ F[t]
pv(A)(v) = 0 . pv(t) A mA(t)
mA(t) = q(t)pv(t) + r(t) 0 ≤ deg r(t) < deg pv(t) .
A
mA(A) = q(A)pv(A) + r(A)
A mA(A) v
0 = mA(A)(v) = q(A)pv(A)(v) + r(A)(v) = r(A)(v) ,
pv(t)
r(t) ≡ 0 , ✷
F V V = {v1, . . . , vn} A ∈ L(V )
vi ∈ V pi(t) ∈ F[t]
pi(A)(vi) = 0 . A mA(t)
p1(t), . . . , pn(t)
k(t) pi(t) (i = 1 . . . , n)
v ∈ V
v = ε1v1 + · · · + εnvn ,
k(A)(v) = ε1k(A)(v1) + · · · + εnk(A)(vn) = 0 ,
i(= 1, . . . , n)
k(t) = qi(t)pi(t) ,

k(A)(vi) = qi(A)pi(A)(vi) = 0 .
A k(t) pi(t) mA(t)
k(t) mA(t)
A
mA(t) | k(t)
k(t) = mA(t) , ✷
V 3 V = {v1, v2, v3}
A A(v1) = v1 + 2v2, A(v2) = v1 − v2
A(v3) = −v1 + v3 A
v1
A
A 2 (v1) = A(v1) + 2A(v2) = v1 + 2v2 + 2(v1 − v2) = 3v1 ,
p1(t) = t 2 − 3 .
V A, A 2 p1(A) = A 2 − 3I
⎡
A = ⎣
1 1 −1
2 −1 0
0 0 1
⎤
⎡
⎦ A 2 = ⎣
⎡
A 2 − 3E = ⎣
0 0 −2
0 0 −2
0 0 −2
3 0 −2
0 3 −2
0 0 1
p2(t) = p1(t) p1(A)(v3) = 0 .
A 2 (v3) = −2v1 − 2v2 + v3.
A 3 (v3) = −2A(v1) − 2A(v2) + A(v3) =
−2(v1 + 2v2) − 2(v1 − v2) + (−v1 + v3) = −5v1 − 2v2 + v3 .
V p3(t)
v3
Av3 A 2 v3 A 3 v3
v1 −1 −2 −5
v2 0 −2
v3 1 1 1
→
⎤
⎦
Av3 A 3 v3
v1 −3
A 2 (v3) 0 1
v3 1 0
→
⎤
⎦
A 3 v3
A(v3) 3
A 2 (v3) 1
v3
−3

A 3 (v3) = A 2 (v3) + 3A(v3) − 3v3 ,
A 3 (v3) − A 2 (v3) − 3A(v3) + 3v3 ,
p3(t) = t 3 − t 2 − 3t + 3 = (t 2 − 3)(t − 1) .
p3(t)
A
v3
V A ∈
L(V ) A
A n V
M0, M1, . . . , Mn A V
dim Mi = i, (i = 0, 1, . . . , n),
{0} = M0 ⊂ M1 ⊂ . . . ⊂ Mn−1 ⊂ Mn = V .
V V
0, 1, n−1
dim V = n. V
A
V ′ 1 A ′ M ′ M ′
V n − 1 Mn−1 = M ′◦ .
Mn−1 A A Mn−1

Mn−1
A M0, M1, . . . , Mn−2
dim Mi = i (i = 0, 1, . . . , n − 2),
M0 ⊂ M1 ⊂ . . . ⊂ Mn−2 ⊂ Mn−1. Mn = V
A
V x1 M1
M1 ⊂ M2 x2 ∈ M2
{x1, x2} M2 x3 ∈ M3 {x1, x2, x3}
M3 V = {x1, x2, . . . , xn}
V i (1 ≤ i ≤ n) {x1, . . . , xi}
Mi i Mi A A(xi) ∈
({x1, . . . , xi}). A V
⎡
⎢
AV = ⎢
⎣
α11 α12 α13 . . . α1n
0 α22 α23 . . . α2n
0 0 α33 . . . α3n
0 0 0 . . . αnn
i (1 ≤ i ≤ n) A − αiiI
Mi Mi−1
A A αii
A p(t)
m p(A) B
A ker p m (A) m
B m W
v W B m−1 (v) = 0 ,
{B m−1 (v), . . . , B(v), v}
B W1 W W = (v, B) ⊕ W1 ,
W
W v
B m−1 (v) = 0 , B m − 1
⎤
⎥
⎦

{Bm−1 (v), . . . , B(v), v}
m−1
i=0
αiB i (v) = 0 .
αj (0 ≤ j ≤ m − 1)
B m−j−1
m−1
αiB i
(v) = αjB m−1 (v) = 0 ,
i=0
B m−1 (v) = 0 α0 = α1 = . . . = αm−1 = 0
{B m−1 (v), . . . , B(v), v}
B
m = 1 , W v
W W1 B
W = (v) ⊕ W1 . m − 1
(B) W B
B m−1
W0 B (B) = (B(v), B)⊕W0 .
(B)
{B m−1 (v), . . . , B(v)}
W 0 = {w ∈ W | B(w) ∈ W0} .
W 0 W B
(v, B) ∪ W 0 W
x ∈ W B(x) ∈ (B) ,
B(x) =
m−1
i=1
B(x) = B
αiB i (v) + w, w ∈ W0
m−2
i=0
αi+1B i (v)
B
x −
m−2
i=0
αi+1B i (v)
W 0
m−2
x −
i=0
+ w
= w .
αi+1B i (v) ∈ W 0 ,

m−2
i=0 αi+1B i (v) ∈ (v, B) , W x
(v, B) W 0
(v, B) ∩ W 0
W 0 W1
x ∈ (v, B) ∩ W0, B(x) ∈ (B(v), B) ∩ W0 ,
B(x) = 0 . x B m−1 (v)
x ∈ (B(v), B) ∩ W0 = {0}
x = 0 .
(v, B) ∩ W0 = {0}.
(v, B) ∩ W 0 W0 W 0 W 0
(v, B) ∩ W 0 V W0 W
w1, . . . , wℓ
W1 = (W ∪ {w1, . . . , wℓ}) .
W = (v, B) ⊕ W1
W1 B
W0 ⊆ W1 ⊆ W 0 , W1 B
W0 B
˜v ∈ W
(˜v, B) m W = (˜v, B) ⊕ W1
B W1 W1
(v, B) (˜v, B) m
dim W1 dim W1
W1
B m1(≤ m)
W1 = (v1, B)⊕W2 W2
W3
Wr B (vr, B)
B m W
(m ≥)m1 ≥ . . . ≥ mr
v0, v1, . . . , vr ∈ W
Bm−1 (v0), . . . , B(v0), v0
Bm1−1 (v1), . . . , B(v1), v1
B mr−1 (vr), . . . , B(vr), vr

W
W = (v0, B) ⊕ (v1, B) ⊕ · · · ⊕ (vr, B) .
(m ≥)m1 ≥ . . . ≥ mr
W B
B B
m−1, m1 −1, . . . , mr −1 1
0
V F
F[t]
A ∈ L(V )
mA(t) = (t − λ1) m1 · · · · · (t − λr) mr
λ1, . . . , λr ∈ F .
V = ker ((A − λ1I) m1 ) ⊕ · · · ⊕ ker ((A − λrI) mr )
A − λiI
Wi = ker((A − λiI) mi ) mi
i(= 1, . . . , r) (mi0 ≥)mi1 ≥
. . . ≥ mis vi0, vi1, . . . , vis ∈ Wi
(A − λiI) mi0−1 (vi0), . . . , (A − λiI)(vi0), vi0
(A − λiI) mi1−1 (vi1), . . . , (A − λiI)(vi1), vi1
(A − λiI) mis−1 (vis),
Wi
. . . , (A − λiI)(vis), vis
Wi = (vi0, (A − λiI)) ⊕ (vi1, (A − λiI)) ⊕ · · · ⊕ (vis, (A − λiI)) .
(vij, (A − λiI)) (j = 0, . . . , s)
A − λiI A
A Wi Ai
j(= 0, . . . , s)
=
A((A − λiI) k )(vij) =
((A − λiI) k+1 )(vij) + λi((A − λiI) k (vij) 0 ≤ k < mij − 1,
λi((A − λiI) k (vij) k = mij − 1

(A − λiI) mi0−1 (vi0), (A − λiI) mi1−1 (vi1), . . . , (A − λiI) mis−1 (vis)
A λi ker(A − λiI)
Ai λi
mi0 − 1, mi1 − 1, . . . , mis − 1
0
⎡
λi
⎢
0
⎢ ⎢
0
⎢
0
⎢
0
⎢
0
⎢ ⎢
Ai = ⎢ ⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎢ ⎢
⎣ 0
1
λi
0
0
0
0
0
0
0
0
0
0
1
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
λi
0
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
0
0
1
λi
0
0
0
0
0
0
0
0
0
0
0
λi
0
0
0
0
0
0
0
0
0
0
1
λi
0
0
0
0
0
· · ·
· · ·
· · ·
· · ·
0
1
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
λi
0
· · ·
· · ·
· · ·
0
0
0
0
0
0
1
λi
0
0
0
0
0
0
0
0
0
0
0
λi
0
0
0
0
0
0
0
0
0
0
1
λi
0
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
0
1
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
λi
0
0
0
0
0
0
0
0
0
0
1
0 0 · · · · · · 0 0 0 · · · · · · 0 0 0 · · · 0 λi
V (i = 1, . . . , r) Wi
A A
Ai (i = 1, . . . , r)
λ1, . . . , λr λi dim ker((A − λiI) mi ) =
mi0 + mi1 + · · · + mis
A
⎤
⎥ .
⎥
⎦

3 W A
W = {w1, w2, w3}
⎡
AW = ⎣
0 −3 −2
1 −1 1
−1 3 1
A w1 0
A p1(A) = A2 + 2A + I.
w2 p2(A) = A2 − A − 2I, w3 p3(A) = A3 − 3A − 2I
0 A
p1(t) = (t + 1) 2 , p2(t) = (t + 1)(t − 2) p3(t) = (t + 1) 2 (t − 2)
mA(t) = (t + 1) 2 (t − 2).
W = ker (A + I) 2 ⊕ ker(A − 2I). A + I
W1 = ker (A + I) 2 2
(A + I) · x = 0 1
s1 =
⎡
⎣ −1
−1
1
v A + I s1
(A + I) · x = s1
xW =
⎡
⎣ ξ1
ξ2
ξ3
⎤
⎦ =
⎡
⎣ −1
0
0
⎤
⎦ +
⎤
⎦ ,
⎡
⎣ −1
−1
1
⎤
⎦
⎤
⎦ · τ, τ ∈ C.
v = −w1. {s1 = (A + I)(v), v}
W1 = ker (A + I) 2
(A − 2I) · x = 0
ker(A − 2I) = {x | x = τ(−w1 + w3), τ ∈ C}
1 −w1 + w3 2
J = {(A + I)(v), v, −w1 + w3} W
(A(A + I))(v) = (A + I) 2 (v) − (I(A + I))(v) = −(A + I)(v),
A(v) = (A + I)(v) − v,

A(−w1 + w3) = 2(−w1 + w3),
A J
⎡
−1
AJ = ⎣ 0
1
−1
0
0
⎤
⎦ .
0 0 2
(A + I)(v) = s1 = −w1 − w2 + w3 v = −w1, W J
B
⎡
BW = ⎣
−1 −1 −1
−1 0 0
1 0 1
AW BW
AJ = B −1
W AWBW.
A
i(= 1, . . . , r) mi0 = 1 A
(t − λi)
ker(A − λiI) 1
(vi0, A − λiI) = (vi0), . . . ,(vis, A − λiI) = (vis)
A
V F A ∈ L(V ) A
A
F[t]
⎤
⎦ ,

n
n

n
n

k

k