1lÙ 5C
§ 1 5m�ÄCIC
1. � V n5m, η1, η2, · · · , ηn V �Ä.
α1 = η1 + η2 + · · ·+ ηn, α2 = η2 + · · ·+ ηn, · · · , αn = ηn
(1) y²: α1, α2, · · · , αn V �Ä;
(2) ¦dÄ η1, η2, · · · , ηn �Ä α1, α2, · · · , αn �LÞÝ
;
(3) � α3Ä η1, η2, · · · , ηn e�I (a1, a2, · · · , an), ¦ α3Ä α1, α2,
· · · , αn e�I.
): (1), (2) Ï
(α1, α2, · · · , αn) = (η1, η2, · · · , ηn)
1 0 · · · 0
1 1 · · · 0
...
...
. . .
...
1 1 · · · 1
,
�
T =
1 0 · · · 0
1 1 · · · 0
...
...
. . .
...
1 1 · · · 1
,
K T _, l
α1, α2, · · · , αn V �Ä,
dÄ η1, η2, · · · , ηn �Ä α1, α2,
· · · , αn �LÞÝ
T .
(3) �
α = (η1, η2, · · · , ηn)
a1
...
an
,
§ 1 5m�ÄCIC · 39 ·
K
α = (α1, α2, · · · , αn)A−1
a1
...
an
,
¤± α3Ä α1, α2, · · · , αn e�I (a1, a2 − a1, a3 − a2, · · · , an − an−1).
2. 3 K4 ¥, ¦dÄ ξ1, ξ2, ξ3, ξ4 �Ä η1, η2, η3, η4 �LÞÝ
, ¿¦þ
α3½Äe�I.
(1) ξ1 = (1, 0, 0, 0), ξ2 = (0, 1, 0, 0), ξ3 = (0, 0, 1, 0), ξ4 = (0, 0, 0, 1);
η1 = (2, 1,−1, 1), η2 = (0,−1, 1, 0), η3 = (−1,−1, 2, 1), η4 = (2, 1, 1, 3);
α = (x1, x2, x3, x4)3 η1, η2, η3, η4 e�I;
(2) ξ1=(1, 2,−1, 0), ξ2=(1,−1, 1, 1), ξ3=(−1, 2, 1, 1), ξ4=(−1,−1, 0, 1);
η1 = (2, 1, 0, 1), η2 = (0, 1, 2, 2), η3 = (−3,−1,−1, 1), η4 = (1, 3, 1, 2);
α = (1, 0, 0, 0)3 ξ1, ξ2, ξ3, ξ4 e�I;
(3) ξ1=(1, 1, 1, 1), ξ2=(1, 1,−1,−1), ξ3=(1,−1, 1,−1), ξ4=(1,−1,−1, 1);
η1 = (1, 1, 0, 1), η2 = (2, 1, 2, 1), η3 = (1, 1, 1, 0), η4 = (0, 1,−1,−1);
α = (1, 0, 0,−1) 3 η1, η2, η3, η4 e�I.
): (1) T =
2 0 −1 2
1 −1 −1 1
−1 1 2 1
1 0 1 3
,
y1
y2
y3
y4
= T−1
x1
x2
x3
x4
= 12
−1 −5 −5 4
2 0 2 −2
−2 −4 −4 4
1 3 3 −2
x1
x2
x3
x4
=
1
2
−x1 − 5x2 − 5x3 + 4x4
2x1 + 2x3 − 2x4
−2x1 − 4x2 − 4x3 + 4x4
x1 + 3x2 + 3x3 − 2x4
.
(2) T =
1 0 0 1
1 1 −1 1
0 1 0 1
0 0 2 0
, α3Ä ξ1, ξ2, ξ3, ξ4 e�I 113
3
5
−2
−3
.
· 40 · 1lÙ 5C
(3) T =
1
4
3 6 3 −1
1 0 1 3
−1 2 1 −1
1 0 −1 −1
, ξ3Äη1, η2, η3, η4 e�I
−2
4
−5
3
.
3. UþK (2), ¦þ, §3Ä ξ1, ξ2, ξ3, ξ4 e�I´3Ä η1, η2, η3, η4
e�I� 2�.
): (0, 4, 2, 6).
4. �K[x]n L«Xê3ê K ¥gê�u n�õª|¤�5m.
fi(x) = (x− a1) · · · (x− ai−1)(x− ai+1) · · · (x− an), i = 1, · · · , n,
Ù¥ ai ∈ K (i = 1, 2, · · · , n)pØÓ�ê.
(1) y²: f1(x), f2(x), · · · , fn(x) |¤ K[x]n �Ä;
(2) � a1, a2, · · · , an �N n gü 1, ε1, · · · , εn−1, ¦dÄ 1, x, x2,
· · · , xn−1 �Ä f1(x), f2(x), · · · , fn(x)�LÞÝ
.
): (1) y f1(x), f2(x), · · · , fn(x) 5Ã'=. �
k1f1(x) + k2f2(x) + · · ·+ knfn(x) = 0,
©O± x = ai \þª, �
kifi(ai) = 0.
Ï fi(ai) 6= 0, ¤± ki = 0, i = 1, 2, · · · , n. � f1(x), f2(x), · · · , fn(x) 5
Ã'. qÏ dimK[x]n = n, f1(x), f2(x), · · · , fn(x)K[x]n �Ä.
(2) ��Ü n gü ´ 1, ε1, · · · , εn−1. K
fi(x) =
xn − 1
x− εi =
xn − εni
x− εi = x
n−1 + εix
n−2 + ε2ix
n−3 + · · ·+ εn−1i ,
�¤¦LÞÝ
1 εn−11 · · · εn−1n−1
1 εn−21 · · · εn−2n−1
...
...
. . .
...
1 1 · · · 1
.
5. ½Â�Xêõª
〈x〉0 = 1, 〈x〉 = x, 〈x〉k = x(x− 1)(x− 2) · · · (x− k + 1), k > 1
(1) ¦K[x]5¥dÄ 1, 〈x〉, 〈x〉2, 〈x〉3, 〈x〉4�Ä 1, x, x2, x3, x4�LÞÝ
;
§ 1 5m�ÄCIC · 41 ·
(2) ¦K[x]5¥õª f(x) = 1+x+x
2+x3+x43Ä 1, 〈x〉, 〈x〉2, 〈x〉3, 〈x〉4
e�I;
∗(3) y²:
n∑
x=0
〈x〉k = 1
k + 1
〈n+ 1〉k+1;
∗(4) dd�Ñê� Dn =
n∑
k=0
k4 �Ïúª.
): (1) 1 = 1
x = 〈x〉
x2 = 0 + x+ x(x− 1) = 0 + 〈x〉+ 〈x〉2
x3 = x+ 3x(x− 1) + x(x− 1)(x− 2) = 〈x〉+ 3〈x〉2 + 〈x〉3
x4 = 〈x〉+ 7〈x〉2 + 6〈x〉3 + 〈x〉4
�¤¦LÞÝ
T =
1 0 0 0 0
0 1 1 1 1
0 0 1 3 7
0 0 0 1 6
0 0 0 0 1
.
(2) (1, 4, 11, 7, 1).
(3) ´ 〈x+ 1〉k+1 − 〈x〉k+1 = (k + 1)〈x〉k. ¤±
n∑
x=0
〈x〉k = 1
k + 1
n∑
x=0
[〈x+ 1〉k+1 − 〈x〉k+1]
=
1
k + 1
[
n+1∑
x=1
〈x〉k+1 −
n∑
x=0
〈x〉k+1
]
=
1
k + 1
(〈n+ 1〉k+1 − 〈0〉k+1)
=
1
k + 1
〈n+ 1〉k+1.
(4) Ï x4 = 〈x〉+ 7〈x〉2 + 6〈x〉3 + 〈x〉4, ¤±
Dn =
n∑
x=0
x4 =
n∑
x=0
(〈x〉+ 7〈x〉2 + 6〈x〉3 + 〈x〉4)
=
1
2
〈n+ 1〉2 + 7
3
〈n+ 1〉3 + 6
4
〈n+ 1〉4 + 1
5
〈n+ 1〉5
=
1
30
n(n+ 1)(2n + 1)(3n2 + 3n− 1).
· 42 · 1lÙ 5C
§ 2 ÄCé5CÝ
�K
1. ½ K3�üÄ:
ξ1 = (1, 1,−1),
ξ2 = (1, 0,−1),
ξ3 = (1, 1, 1),
η1 = (1,−1, 2),
η2 = (2,−1, 2),
η3 = (−2, 1, 1).
� A K3 �5C, ¦:
A ξi = ηi i = 1, 2, 3.
(1) ¦dÄ ξ1, ξ2, ξ3 �Ä η1, η2, η3 �LÞÝ
;
(2) ¦ A 3Ä ξ1, ξ2, ξ3 e�Ý
;
(3) ¦ A 3Ä η1, η2, η3 e�Ý
;
(4) � α = (2,−1, 3), ©O¦ A α3Ä ξ1, ξ2, ξ3 Ä η1, η2, η3 e�I.
): � K3IOÄ ε1 = (1, 0, 0), ε2 = (0, 1, 0), ε3 = (0, 0, 1), -
B =
1 1 11 0 1
−1 −1 1
, C =
1 2 −2−1 −1 1
2 2 1
,
Kk
(ξ1, ξ2, ξ3) = (ε1, ε2, ε3)B, (η1, η2, η3) = (ε1, ε2, ε3)C.
(1) du (η1, η2, η3) = (ε1, ε2, ε3)C = (ξ1, ξ2, ξ3)B
−1C, �dÄ ξ1, ξ2, ξ3
�Ä η1, η2, η3 �LÞÝ
T = B−1C =
−5
2
−3 3
2
2 3 −3
3
2
2 −1
2
.
(2) du (A (ξ1),A (ξ2),A (ξ3)) = (η1, η2, η3) = (ξ1, ξ2, ξ3)B
−1C, � A
3Ä ξ1, ξ2, ξ3 e�Ý
A = B−1C =
−5
2
−3 3
2
2 3 −3
3
2
2 −1
2
.
§ 2 ÄCé5CÝ
�K · 43 ·
(3) � A 3Ä η1, η2, η3 e�Ý
A
′, K
A′ = T−1AT = (B−1C)−1(B−1C)(B−1C) = B−1C =
−5
2
−3 3
2
2 3 −3
3
2
2 −1
2
.
(4) α3Ä ξ1, ξ2, ξ3 Ä η1, η2, η3 e�I©O
B−1
2−1
3
C−1
2−1
3
,
Ïd A α3Ä ξ1, ξ2, ξ3 e�I
AB−1
2−1
3
= 1
2
7−11
−1
,
A α3Ä η1, η2, η3 e�I
A′C−1
2−1
3
= 1
2
−76
5
.
2. � A ∼ C, B ∼ D, y²:(
A 0
0 B
)
∼
(
C 0
0 D
)
.
y²: 3_Ý
T1, T2, ¦�
T−11 AT1 = C, T
−1
2 BT2 = D,
Ïd T =
(
T1 0
0 T2
)
_,
T−1
(
A 0
0 B
)
T =
(
T−11 AT1 0
0 T−12 BT2
)
=
(
C 0
0 D
)
.
¤± (
A 0
0 B
)
∼
(
C 0
0 D
)
.
· 44 · 1lÙ 5C
3. � A _, y²: AB BA q.
y²: du A−1(AB)A = BA, � AB ∼ BA.
4. � A _,
A ∼ B, y²: B _,
A−1 ∼ B−1.
y²: du T,A�_, ¤± B _,
B−1 = (T−1AT )−1 = T−1A−1T,
� A−1 ∼ B−1.
5. � A ∼ B, y²: AT ∼ BT.
y²: 3_Ý
T , ¦� T−1AT = B. � BT = (T−1AT )T =
TTATT−T.
6. � A ∼ B, f(x) ∈ K[x], y²: f(A) ∼ f(B).
y²: 3_Ý
T , ¦� T−1AT = B. �
T−1(f(A))T = f(T−1AT ) = f(B).
7. y²:
λ1
λ2
. . .
λn
∼
λi1
λi2
. . .
λin
,
Ù¥ (i1, i2, · · · , in)´ (1, 2, · · · , n)�ü�.
y²: � V ´ n5m, ε1, · · · , εn ´ V �Ä. A V �5C,
½Â
A εi = λiεi,
K A 3Ä ε1, · · · , εn e�Ý
A =
λ1
λ2
. . .
λn
.
du (i1, i2, · · · , in) ´ (1, 2, · · · , n) �ü�, Ïd εi1 , · · · , εin E V �
Ä,
A εij = λijεij , j = 1, · · · , n.
§ 2 ÄCé5CÝ
�K · 45 ·
� A 3Ä εi1 , · · · , εin e�Ý
B =
λi1
λi2
. . .
λin
,
l
A ∼ B.
8. � x, y, z ∈ K , -
A =
x y zy z x
z x y
, B =
z x yx y z
y z x
, C =
y z xz x y
x y z
.
y²: A,B,C *dq.
y²: �Ý
P =
0 1 00 0 1
1 0 0
, Q =
0 0 11 0 0
0 1 0
,
K P,Q�_,
P−1AP = C, Q−1AQ = B,
¤± A ∼ B, A ∼ C. dq'X�D45, � B ∼ C.
∗9. y²:
1 1 · · · 1
1 1 · · · 1
...
...
. . .
...
1 1 · · · 1
n
∼
n 0 · · · 0
0 0 · · · 0
...
...
. . .
...
0 0 · · · 0
.
y²: � V ´ n5m, ε1, · · · , εn ´ V �Ä. A V �5C,
½Â
A εi = ε1 + ε2 + · · ·+ εn, i = 1, 2, · · · , n.
K A 3Ä ε1, · · · , εn e�Ý
A =
1 1 · · · 1
1 1 · · · 1
...
...
. . .
...
1 1 · · · 1
n
.
· 46 · 1lÙ 5C
q´
α1 = ε1 + ε2 + · · ·+ εn, α2 = ε1 − ε2, · · · , αn = ε1 − εn
E V �Ä,
A 3Ä α1, · · · , αn e�Ý
B =
n 0 · · · 0
0 0 · · · 0
...
...
. . .
...
0 0 · · · 0
.
l
A ∼ B.
§ 3 5C�A�A�þ
1. ¦Eêþ5m V �5C A �A�A�þ, � A 3
V �Äe�Ý
´:
(1) A =
(
2 5
4 3
)
; (2) A =
(
0 a
−a 0
)
(a 6= 0);
(3) A =
1 1 1 1
1 1 −1 −1
1 −1 1 −1
1 −1 −1 1
; (4) A =
5 6 −3−1 0 1
1 2 −1
;
(5) A =
0 0 10 1 0
1 0 0
; (6) A =
0 −2 −1−2 0 −1
−1 −3 1
;
(7) A =
4 3 0−3 −2 0
2 −6 2
.
): êiL«A�, ;�3��þÒ´A�A�þ.
(1) 7, (1, 1); −2, (5,−4).
(2) ai, (1, i); −ai, (i, 1).
(3) 2, k(1, 1, 0, 0) + l(1, 0, 1, 0) +m(1, 0, 0, 1); −2, (−1, 1, 1, 1).
(4) 2, (−2, 1, 0); 1 +√3, (−3, 1,−2 +√3); 1−√3, (−3, 1,−2 −√3).
(5) 1, k(1, 0, 1) + l(0, 1, 0); −1, (−1, 0, 1).
(6) −3, (1, 1, 1); 2, (1, 1,−4).
§ 3 5C�A�A�þ · 47 ·
(7) 2, (0, 0, 1); 1, (−1, 1, 8).
2. y²: îAp�m���C�A� (Xk�{)U´ ±1.
y²: � α´áu��C A �A� λ0 �A�þ, K
0 6= (α,α) = (A α,A α) = λ20(α,α),
Ïd λ20 = 1, λ0 = ±1.
3. y²: "Ý
(,�u"�Ý
) �A��".
y²: � α´áu"Ý
A�A� λ0 �A�þ, K Aα = λ0α. d
u
0 = Akα = λk0α,
� λk0 = 0, λ0 = 0.
4. � A = (a1, a2, · · · , an) ∈ Rn (ai Ø�"), ¦Ý
ATA�A�
A�þ.
): � ai 6= 0. A� 0éA�A�þ´
αj = (0, · · · , 0,ai, · · · ,−aj, 0, · · · , 0)
j i
, (j=1, · · · , i−1, i+1, · · · , n)
�5|Ü. A�
n∑
j=1
a2j�A�þ´(a1 , · · · , an).
5. � A ∈Mn(K). y²: 3K þ�gêØL n2�õª f(x),
¦ f(A) = 0.
y²: ÏMn(K) ´ K þ n
2 5m, � E,A,A2, · · · , An2−1, An2
5'. u´3Ø�"� ai ∈ K, i = 1, · · · , n2 ¦�
a0E + a1A+ · · ·+ an2−1An2−1 + an2An2 = 0.
-
f(x) = an2x
n2 + an2−1x
n2−1 + · · ·+ a1x+ a0,
K f(A) = 0.
∗6. � A = (aij) ∈Mn(K), fª∣∣∣∣∣∣∣∣∣∣
ai1i1 ai1i2 · · · ai1ik
ai2i1 ai2i2 · · · ai2ik
...
...
. . .
...
aiki1 aiki2 · · · aikik
∣∣∣∣∣∣∣∣∣∣
, (1 6 i1 < i2 < · · · < ik 6 n),
· 48 · 1lÙ 5C
¡ A� k�Ìfª. -A�õª
χA(λ) = |λE −A| = λn − a1λn−1 + · · · + (−1)n−1an−1λ+ (−1)nan.
y²: ak �u A��Ü k�ÌfªÚ.
y²: r |λE −A|�z�
−a1j
−a2j
...
λ− ajj
...
−anj
Ñ
¤ü�:
0
...
0
λ
0
...
0
−a1j
−a2j
...
−ajj
...
−anj
K1�ª |λE −A| ©) 2n n�1�ªÚ, Ù¥z1�ª��Ñ´
þãü«/ª.
� Ak ?¹k k λ�f1�ª, Ù λ ?u j1, · · · , jk �, ò Ak Uù
k�Ðm, �
Ak = λ
k · (−1)n−kDn−k,
Ù¥ Dn−k 3 A¥y�1 j1, · · · , jk �!1 j1, · · · , jk 1
��� n − k �
Ìfª. �ù k λ�H n�1�ª¥¤kU� k , K Dn−k Ò�H
¤k Ckn Ìfª. l
χA(λ)¥ λ
n−k �Xê�u (−1)k ¦± A�¤k k�
ÌfªÚ. Ïd ak A�¤k k�ÌfªÚ.
∗7. y²: AB BA kÓ�A�.
y²: âSK 5–8�1 4K, � |A| 6= 0, AC = CA, k∣∣∣∣∣ A BC D
∣∣∣∣∣ = |AD −CB|.
§ 3 5C�A�A�þ · 49 ·
Ïd ∣∣∣∣∣ λE BA E
∣∣∣∣∣ = |λE −AB|.
qÏ (
0 E
E 0
)(
λE B
A E
)(
0 E
E 0
)
=
(
E A
B λE
)
,
ü>�1�ª, =�∣∣∣∣∣ E AB λE
∣∣∣∣∣ = |λE −BA| =
∣∣∣∣∣ λE BA E
∣∣∣∣∣ = |λE −AB|.
Ïd AB BA kÓ�A�õª, l
kÓ�A�.
∗8. � A ∈ Mn(C). y²: 3_Ý
T ∈ GL(n,C), ¦ T−1AT þ
n�Ý
.
y²: é n ^êÆ8B{. � n = 1(Øg,¤á. y�(Øé n − 1
�Ý
¤á.
� λ1 ´ A�A�, A�A�þ´ α1 ∈ Cn. r α1 *¿¤ Cn
�Ä α1, α2, · · · , αn. - T1 = (α1, α2, · · · , αn), K T1 _,
AT1 = T1
(
λ1 ∗
0 A1
)
, = T−11 AT1 =
(
λ1 ∗
0 A1
)
,
Ù¥ A1 ∈Mn−1(C). d8Bb�, 3_Ý
T2 ∈Mn−1(C), ¦�
T−12 A1T2 =
λ2 ∗
. . .
0 λn
,
- T = T1
(
1 0
0 T2
)
, K T _,
T−1AT =
λ1 ∗
. . .
0 λn
.
∗9. � A ∈ Mn(C), f(x)EXêõª. y²: XJ A��ÜA�
λ1, λ2, · · · , λn, K f(A)��ÜA� f(λ1), f(λ2), · · · , f(λn).
· 50 · 1lÙ 5C
y²: dSK 8, 3_Ý
T , ¦
T−1AT =
λ1 ∗
λ2
. . .
0 λn
,
ùp λ1, · · · , λn ´ A��ÜA�. l
T−1f(A)T = f(T−1AT ) = f
λ1 ∗
λ2
. . .
0 λn
=
f(λ1) ∗
f(λ2)
. . .
0 f(λn)
.
¤± f(λ1), f(λ2), · · · , f(λn) f(A)��ÜA�.
§ 4 é�z5C
1. SK 8–3 1K¥�Ý
, =
´±é�z�? 3é�z�¹
e, ¦ÑA�LÞÝ
Úé�Ý
.
): (1) T =
(
1 5
1 −4
)
, T−1AT = diag(7,−2).
(2) T =
(
1 i
i 1
)
, T−1AT = diag(ai,−ai).
(3) T =
−1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
, T−1AT = diag(−2, 2, 2, 2).
(4) T =
−2 −3 −31 1 1
0 −2 +√3 −2−√3
, T−1AT = diag(2, 1 + √3, 1 −
√
3).
§ 4 é�z5C · 51 ·
(5) T =
−1 1 00 0 1
1 1 0
, T−1AT = diag(−1, 1, 1).
(6), (7) Øé�z.
2. 3K[x]n ¥, ¦©C D :
D(f(x)) = f ′(x)
�A�õª, ¿y²: D 3?ÛÄe�Ý
ÑØU´é�Ý
.
): � K[x]n �Ä 1, x, x
2, · · · , xn−1. K D 3ùÄe�Ý
´
D =
0 1 0 · · · 0
0 0 2 · · · 0
...
...
...
. . .
...
0 0 0 · · · 0
.
l
D �A�õª χD(λ) = λ
n. XJ D é�z, K3_Ý
T ¦
� T−1AT = 0, = D = 0,
D Ø´"C, gñ.
3. � A ê K þ n5m V �5C, ÷v A 2 = A . ¦ A
�A�, ¿y² A é�z.
y²: �
V1 = {A α | α ∈ V }, V2 = {α−A α | α ∈ V }.
Ké?¿� α ∈ V , k α = A α+ (α−A α), Ïd
V = V1 + V2.
e α ∈ V1 ∩ V2, K
α = A β = γ −A γ, β, γ ∈ V.
u´
A α = A 2β = A β = α, (∗)
A α = A γ −A 2γ = A γ −A γ = 0. (∗∗)
¤± α = A α = 0, = V1 ∩ V2 = 0. ù�Òy²
V = V1 ⊕ V2.
· 52 · 1lÙ 5C
éu V1 ¥�þ α, k (*)ª¤á, `² V1 ´áuA� 1�A�fm. a
q/d (**)ª� V2 ´áuA� 0�A�fm. âíØ 4.5, A é
�z. A �A� 0, 1.
(5: |^SK 9�{\±y²)
4. �
A =
(
1 1
1 0
)
.
(1) kòÝ
Aé�z, 2¦ An;
∗(2) |^þã(J, ¦ÜÅ@êê� (ë18ÙöS 6–1.8)�Ïúª.
): (1) A�A�
1±√5
2
, A�A�þ
1 +
√
5
2
1
,
1−
√
5
2
1
.
-
T =
1 +
√
5
2
1−√5
2
1 1
,
K
T−1AT =
1 +
√
5
2
0
0
1−√5
2
.
l
An = T
(
1 +
√
5
2
)n
0
0
(
1−√5
2
)n
T−1
=
1√
5
1 +
√
5
2
1−√5
2
1 1
(
1 +
√
5
2
)n
0
0
(
1−√5
2
)n
·
1 −
1−√5
2
−1 1 +
√
5
2
= √5
5
(
αn+1 − βn+1 αn − βn
αn − βn αn−1 − βn−1
)
,
§ 4 é�z5C · 53 ·
Ù¥ α =
1 +
√
5
2
, β =
1−√5
2
.
(2) -
α0 =
(
1
0
)
, α1 =
(
1
1
)
, · · · , αn =
(
an+1
an
)
,
Ù¥ ai ´1 i ÜÅ@êê. K
αn = A
nα0.
¤± (
an+1
an
)
= Anα0 =
√
5
5
(
αn+1 − βn+1 αn − βn
αn − βn αn−1 − βn−1
)(
1
0
)
,
�
an =
√
5
5
(αn − βn) =
√
5
5
[(
1 +
√
5
2
)n
−
(
1−√5
2
)n]
.
5. � A = (aij) n�þn�Ý
. y²:
(1) e aii 6= ajj (i 6= j), K A é�z;
(2) e a11 = a22 = · · · = ann,
�k aij 6= 0 (i 6= j), K AØé
�z.
y²: (1) du A´þn�Ý
, � a11, a22, · · · , ann A� n A�.
e� i 6= j aii 6= ajj , K A k n ØÓ�A�, l
A é�z.
(2) (y) ® A�A� λ0 = a11 = · · · = ann, X A é�z, K
3_Ý
T , ¦�
T−1AT = diag(λ0, · · · , λ0).
u´
A = T
λ0
. . .
λ0
T−1 =
λ0
. . .
λ0
.
= AXþ
, b�gñ.
6. �k©¬é�Ý
A =
A1 0 · · · 0
0 A2 · · · 0
...
...
. . .
...
0 0 · · · As
.
· 54 · 1lÙ 5C
y²: A é�z�¿©7^´z Ai �é�z.
y²: ¿©5´w,�. ey75.
� A é�z, K3_Ý
T , ¦�
T−1AT =
λ1
λ2
. . .
λn
.
u´
AT = T
λ1
λ2
. . .
λn
.
-
T =
T1
T2
...
Ts
,
Ù¥ Ti �1ê�u Ai ��ê ri. K
AiTi = Ti
λ1
λ2
. . .
λn
.
ù`² Ti �"�þÑ´ Ai �A�þ. qÏ T _, � T �1þ5
Ã', Ï
Ti �1þ5Ã'. u´ rankTi = Ai ��ê, Ti ���
u Ai ��ê. Ïd Ai k ri 5Ã'�A�þ, `² Ai �é�z.
7. � λ1, λ2´5C A �üØÓ�A�, ε1, ε2©O´ A �áu
A� λ1, λ2 �A�þ. y²: ε1 + ε2 Ø´ A �A�þ.
y²: (y) XJ ε1 + ε2 ´ A �áu,A� λ0 �A�þ, K
A (ε1 + ε2) = λ0(ε1 + ε2).
q A (ε1 + ε2) = A ε1 + A ε2 = λ1ε1 + λ2ε2, ¤±
(λ1 − λ0)ε1 + (λ2 − λ0)ε2 = 0.
§ 4 é�z5C · 55 ·
d λ1 6= λ2 � ε1, ε2 5Ã', Ïd
λ1 − λ0 = 0, λ2 − λ0 = 0,
�� λ1 = λ0 = λ2, gñ.
8. y²: XJ5C A ±z"þ§�A�þ, K A I
þ¦ÈC.
y²: �é,"þ α k A α = kα, é,"þ β, k
A β = mβ. XJ k 6= m, KâSK 7�(Ø, α+ β Ø´ A �A�þ. X
J α+ β = 0, Kk A β = −A α = −kα = kβ, k 6= m gñ. Ïd α+ β ´
"þ, K�gñ.
∗9. � A ∈ Mn(K), y²: XJ rankA+ rank(A − E) = n, K A é�
z.
y²: dSK 5–8.13 , rankA + rank(A − E) = n �¿©7^´
A2 = A. =é A�?�þ α k Aα = α. q A(A − E) = 0, �é A− E
�?�þ β k Aβ = 0.
� A��þ|�4Ã'| α1, · · · , αr, A− E �4Ã'�þ|
β1, · · · , βn−r (Ï rankA + rank(A − E) = n). ey α1, · · · , αr, β1, · · · ,
βn−r 5Ã'.
�k
r∑
i=1
kiαi +
n−r∑
j=1
mjβj = 0.
K
r∑
i=1
kiAαi +
n−r∑
j=1
mjAβj =
r∑
i=1
kiAαi =
r∑
i=1
kiαi = 0.
u´ k1 = · · · = kr = 0, ?
m1 = · · · = mn−r = 0. ¤± α1, · · · , αr, β1, · · · ,
βn−r 5Ã'.
- T = (α1, · · · , αr, β1, · · · , βn−r), K T _,
AT = A(α1, · · · , αr, β1, · · · , βn−r) = (α1, · · · , αr, β1, · · · , βn−r)
(
Er 0
0 0
)
,
=
AT = T
(
Er 0
0 0
)
,
· 56 · 1lÙ 5C
l
T−1AT =
(
Er 0
0 0
)
.
∗10. � A ∈ Mn(K), y²: XJ rank(A + E) + rank(A − E) = n, K A
é�z.
y²: þKaq, Ñ.
∗11. � A,B ∈ Mn(K),
AB = BA. y²: XJ A, B Ñé�z, K
3_Ý
T ∈Mn(K), ¦ T−1AT T−1BT Óé�Ý
.
y²: � A�ØÓA� λ1, λ2, · · · , λs, Ù¥ λi �ê ri. du A
é�z, 3_Ý
T , ¦
T−1AT = A1 =
λ1Er1
. . .
λsErs
.
- B1 = T
−1BT , K A1B1 = B1A1.
- B1 = (Bij), Ù©¬ª A1 Ó, Kd A1B1 = B1A1 �
λiBij = Bijλj
u´� i 6= j k Bij = 0, =
B1 =
B11
B22
. . .
Bss
.
qÏ B é�z, B1 é�z, l
dþKz Bii Ñé�z. =3
_Ý
Si ∈Mri(K)¦
S−1i BiiSi =
λi1
. . .
λiri
.
-
T = T1
S1
. . .
Ss
,
§ 5 5C�ØCfm · 57 ·
K T _,
T−1AT =
S−11
. . .
S−1s
λ1Er1
. . .
λsErs
S1
. . .
Ss
=
λ1Er1
. . .
λsErs
,
T−1BT =
S−11
. . .
S−1s
B11
. . .
Bss
S1
. . .
Ss
=
S−11 B11S1
. . .
S−1s BssSs
=
λ11
. . .
λ1r1
. . .
λs1
. . .
λsrs
Óé�/.
§ 5 5C�ØCfm
1. � A ´5m V �5C, ® A 3Ä η1, η2, · · · , ηn e�Ý
´
A =
0 1
. . .
. . .
. . . 1
0
.
· 58 · 1lÙ 5C
¦ A �¤kØCfm.
): �W ´ A �"ØCfm. Äky²: W 7¹,Äþ
ηi.
�
0 6= α = akηk + · · ·+ anηn ∈W, ak 6= 0.
K A k−1α = akη1 ∈W , η1 ∈W .
2� ηk ´¹u W ¥�eI�Äþ, K
ηk−1 = A ηk, ηk−2 = A
2ηk, · · · , η1 = A k−1ηk ∈W.
e¡y
W = L(η1, · · · , ηk).
^y{. Xk α ∈W , � α 6∈ L(η1, · · · , ηk), K
α = a1η1 + · · · + amηm, Ù¥am 6= 0,m > k.
u´
am−kη1 + · · · + am−1ηk + amηk+1 = A m−k−1α ∈W.
qÏ am−kη1 + · · ·+ am−1ηk ∈W , am 6= 0, � ηk+1 ∈ W , gñ. Ïd A �
¤kØCfm"fm±9 L(η1, · · · , ηk), k = 1, · · · , n.
2. � A îAp�m���C, y²: A �ØCfm���Ö
´ A �ØCfm.
y²: �W ´ A �ØCfm, K A (W ) ⊆ W . du��C7_,
Ïd A (W ) =W .
éu?¿� β ∈W⊥, α ∈W , 73 γ ∈W ¦ α = A γ. u´
(A β, α) = (A β,A γ) = (β, γ) = 0.
ù`² β ∈W⊥, �W⊥´ A �ØCfm.
3. �A ´5m V �5C,W A �ØCfm. y²: A (W )
´ A �ØCfm.
y²: du W ´ A �ØCfm, A (W ) ⊆ W , Ïd A (A (W )) ⊆
A (W ), `² A (W )´ A �ØCfm.
4. � V ´Eêþ� n5m, A , B ´ V �5C,
A B =
BA . y²:
(1) XJ λ´ A �A�, @o, Vλ ´B �ØCfm;
§ 5 5C�ØCfm · 59 ·
(2) A , B �kú��A�þ.
y²: (1) é?¿� α ∈ Vλ,
A (Bα) = B(A α) = B(λα) = λBα.
Ïd Bα ∈ Vλ, `² Vλ ´B �ØCfm.
(2) dþ, Vλ ´B �ØCfm, l
B|Vλ ´ Vλ þ�5C. u´
B|Vλ kA� µ±9A�A�þ β ∈ Vλ, ¦
Bβ = B|Vλ(β) = µβ.
qÏ β ∈ Vλ, A β = λβ. ¤± β ´ A B �ú�A�þ.
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