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2010 � 5 d
§1. ���$ �
�
1 ;,PG S, �PG {Si|i ∈ I}, I 6\BM
�S6 ∼. ._�T (x, y) ∈ S × S, � (x, y) ∈ T , h� x F y \A>�
S6 x ∼ y; � (x, y) 6∈ T , h� x F y |\A>�S6 x 6∼ y.
) � T ,Ur2bA> ∼, h._�T% x, y ∈ S, Ml+\A>�Ml|
\A>�pl
a�R�xa�R�
3 PG S %R92bA> ” ∼ ” �6 S %R9&TA>��Cy��
(1) 4�K�._�T% x ∈ S, \ x ∼ x;
(2) .�K�._�T% x, y ∈ S, � x ∼ y, h y ∼ x;
(3) �*K�._�T% x, y, z ∈ S, � x ∼ y, y ∼ z, h x ∼ z.
�� 1 PG S %R97ld, S %R9&TA>�
&� � {Si | i ∈ I} " S %R97l�,UA> x ∼ y :⇐⇒ x ` y g2R9l
{�?�r��6&TA>�
(1) ._�T% x ∈ S, �g i ∈ I �$ x ∈ Si. x F x gRl{�*SR x ∼ x;
(2) � x ∼ y, R x F y g2Rl{�*S y ` x #_2Rl�W� y ∼ x;
(3) � x ∼ y, y ∼ z, R x F y #_2Rl� y F z #_2Rl�*S x, y, z #
_2Rl�� x ∼ y.
�� 2 PG S %R9&TA> ” ∼ ” d, S %R97l�
&� .�T% x ∈ S, w Sx = {y | y ∈ S, y ∼ x}. ?�Or�6 S %R97
l�
(1) ._�T% x ∈ S, [_ x ∼ x, R x ∈ Sx. � S {�Rb(
#_�Rl�
R S = ⋃Sx.
1
(2) ._�Tpl Sy, Sz, Ml Sy
⋂Sz = ∅, Ml Sy = Sz. !���� Sy⋂Sz 6=
∅. � x ∈ Sy
⋂Sz, h�,U\ x ∼ y, x ∼ z. [.�Ks y ∼ x, x ∼ z, f[�*
K�g$ y ∼ z. � y ∈ Sz. ^R��._�T% a ∈ Sy, \ a ∼ y, y ∼ z. W�
a ∼ z. R Sy ⊆ Sz. 2m Sz ⊆ Sy. � Sy = Sz .
) . S ^J&T7lTUg_�
(1) W. S %N`K6. Si %N`�K�6G�
(2) �C�?j#&TA> ”∼” %,o�i; Si %b(*=\1�+l%b(
*|\%Kz�gSW.l Si %b(Kz%N`�K6. Si %�9b(%N`�
(3) g (2) %
-?�e�:ltU%A>Mle)�xQex�{% ÆbR
g�
*S�:%N`+g_�
(1) Mj S {%b(g&TA>%��q��C��q�?�7�2%l�h�
���q64���q�
(2) Mj:l% Æb�
(3) N`l%9%�
mP1D:l% ÆbN`}Rl�^1N`p9 S.
§2. �Æ �����(� ���
Fm×n {%D'A>� F n×n {%D'A>� F n×n {�/.�bn%G2A
>� Rn×n {%qXD'A>� F �*\CKhU%2>A>�CKhU V %F
q�%&TA>�
4 � A,B ∈ Fm×n, � AD'_ B, ��gg�bn P ∈ Fm×m, Q ∈ F n×n,
�$ A = PBQ.
g Fm×n {%D'A>"&TA>�bn%y"D'A>%4���q�}R
l% Æb"
(
Ir 0
0 0
)
. �k&TA>g76 min{m,n}+ 1 l�
5 � A,B ∈ F n×n, � A D'_ B, ��gg�bn P ∈ F n×n, �$
A = PBP−1.
g F n×n {%D'A>"&TA>�bn%Js�W~���W~��&W~-
"D'A>%4���q�}Rl% Æb"\m
}H (g F n×n {) M Jordan
}H (g Cn×n {).
6 � A,B ∈ F n×n, � A G2_ B, ��gg�bn P ∈ F n×n, �$
A = PBP ′.
g F n×n {%G2A>"&TA>�
g Cn×n %�/.�bn{�bn%y"G2A>%4���q�}Rl% Æ
b"
(
Ir 0
0 0
)
. �kG2A>g76 n+ 1 l�
2
g Rn×n %�/.�bn{�bn%yFqBKw%"G2A>%4���q�
}Rl% Æb"
Ip 0 00 −Iq 0
0 0 0
. �kG2A>g76 (n+1)(n+2)
2
l�
§3. !# 1: �%�'����&�
� 1 � A ∈ Fm×n, B ∈ F n×l, h r(AB) ≤ min{r(A), r(B)}.
&��� � r(A) = r, r(B) = s. A = P
(
Ir 0
0 0
)
Q, B = P1
(
Is 0
0 0
)
Q1.
W QP1 7� 2 × 2 i� QP1 =
(
Cr×s D
E F
)
. r(AB) = r(P
(
Cr×s 0
0 0
)
Q1) =
r(C) ≤ min(r, s).
� 2 r(A+B) ≤ r(A) + r(B).
&��� � r(A) = r, r(B) = s. ��gg�bn P1, P2, Q1, Q2, �$ A =
P1CQ1, B = P2DQ2, mn C =
(
Ir 0
0 0
)
, D =
(
Is 0
0 0
)
. h A + B =
(P1, P2)
(
C 0
0 D
)(
Q1
Q2
)
. *S r(A+B) ≤ r(
(
C 0
0 D
)(
Q1
Q2
)
)≤ r(
(
C 0
0 D
)
)
= r(A) + r(B).
� 3 r(AB) ≥ r(A) + r(B)− n, mn A ∈ Fm×n, B ∈ F n×l.
&��� � r(A) = r, r(B) = s, h�gg�bn P, P1, Q,Q1, �$ A =
P
(
Ir 0
0 0
)
Q, B = P1
(
Is 0
0 0
)
Q1. *S r(AB) = r(
(
Ir 0
0 0
)
QP1
(
Is 0
0 0
)
).
� QP1 =
(
Cr×s D
E F
)
n×n
, r(QP1) = n. h\ r(AB) ≥ r (
(
C 0
0 0
)
) ≥
r(QP1)− (n− r)− (n− s) = r + s− n.
� 4 r(ABC) ≥ r(AB) + r(BC)− r(B).
&��� � r(B) = s, h�gg�bn P,Q, �$ B = P
(
Is 0
0 0
)
Q. *S
ABC = (AP
(
Is
0
)
)(
(
Is 0
)
QC). � r(ABC) ≥ r(AP
(
Is
0
)
) +r(
(
Is 0
)
QC)−
s = r(AP
(
Is 0
0 0
)
Q) + r(P
(
Is 0
0 0
)
QC)− s = r(AB) +r(BC) −r(B).
� 5 (1) � M ∈ Fm×r, � r(M) = r. h.�T A ∈ F r×n, r(MA) = r(A);
3
(2) � N ∈ F r×n, � r(N) = r. h.�T B ∈ Fm×r, r(BN) = r(B);
(3) � M ∈ Fm×r, � r(M) = r. � MX = 0, h X = 0;
(4) � N ∈ F r×n, � r(N) = r. � Y N = 0, h Y = 0.
&��� (1) M = P
(
Ir
0
)
Q = P
(
Q
0
)
= P
(
Q 0
0 Im−r
)(
Ir
0
)
. P1 =
P
(
Q 0
0 Im−r
)
"g�n�r(MA) = r(P1
(
Ir
0
)
A) = r(P1
(
A
0
)
) = r(
(
A
0
)
) =
r(A).
� 6 � A ∈ F n×n.
(1) r� r(A+ I) + r(A− I) ≥ n;
(2) �C r(Ak) = r(Ak+1), h r(Ak+1) = r(Ak+2).
&���
(1) � A ,ov 1 F −1 % %|%7�" r1 F r2, h r1 + r2 ≤ n. 7= A %
Jordan
}H�gs r(A+I) ≥ n−r2, r(A−I) ≥ n−r1,�1 r(A+I)+r(A−I) ≥ n.
(2) K6 Jordan
}H�ex 0 ,ovQ6 0 ,ov.X% Jordan i�
� 7* � A ∈ Rm×n, h r(A′A) = r(A).
&��� (3R)� r(A) = r,h�gg�bn P ∈ Rm×m Qg�bn Q ∈ Rn×n,
�$ A = P
(
Ir 0
0 0
)
Q, �1 A′A = Q′
(
Ir 0
0 0
)
P ′P
(
Ir 0
0 0
)
Q. W P ′P 7
i
(
C1 C2
C3 C4
)
, �{ C1 " r \5n�[ P
′P "q,n�$ C1 q,� r(C1) = r.
W� r(A′A) = r(
(
Ir 0
0 0
)(
C1 C2
C3 C4
)(
Ir 0
0 0
)
) = r
(
C1 0
0 0
)
= r(A).
(32) [. 2, � r(A) = r, h�g B ∈ Rm×r, C ∈ Rr×n, �$ A = BC,
A′A = C ′B′BC. B′B g�� r(B′BC) = r(C). B′BC ∈ Rr×n, C ′ ∈ Rn×r. *S
r(A′A) = r(C ′(B′BC)) = r = r(A).
� 8* � H " n \q,n�r�.�T A ∈ Rn×n, \ r(A′HA) = r(A).
&��� W H q,�*SG2_!8n�R�g n\g�n P , �$ H = P ′P .
r(A′HA) = r(A′P ′PA) = r(PA) = r(A).
§4. !# 2: �%�
�
� 9 (1) � A ∈ Fm×n, � rank(A) = r. Ær� A = A1 + A2 + · · ·+ Ar, mn
rank(Ai) = 1, 1 ≤ i ≤ r;
(2)� A,B ∈ Fm×n,� rank(A) = r, rank(B) = 1. Ær��gg�n P1, P2, · · · , Pr ∈
Fm×m Fg�n Q1, Q2, · · · , Qr ∈ F n×n, �$ A = P1BQ1+P2BQ2+ · · ·+PrBQr.
4
&��� oZD'
}Hr��
� 10 (1) � A ∈ F n×n, r(A) = r. Ær��g B ∈ F n×n � r(B) = n− r, �$
AB = 0;
(2) � A ∈ F n×n, r(A) = r. Ær��g B ∈ F n×n � r(B) = n − r, �$
AB = BA = 0;
(3) � A,B ∈ F n×n, r(A) + r(B) ≤ n. Ær��gg�n M ∈ F n×n, �$
AMB = 0;
(4) (3) {%℄"85R�
� 11 � A ∈ F n×n. Ær�
(1) A = BC = DE, mn C2 = C, D2 = D, B,E " n \g�n�
(1’) A g_DR>sJ% (Ms%) �&�LK6�&n�
(2) A = ST = UV , mn T ′ = T , U ′ = U , S, V " n \g�n�
(2’) A g_DR>sJ% (Ms%) �&�LK6.�n�
� 12
(1) � M ∈ Fm×r, r(M) = r. Ær��gg�n P , �$ PM =
(
Ir
0
)
;
(1’) syybng_DR>s�&J�LK6D'
}H�
(2) � N ∈ F r×n, r(N) = r. Ær��gg�n Q, �$ NQ = (Ir 0);
(2’) Jyybng_DR>s�&s�LK6D'
}H�
(3) � M ∈ Fm×r, r(M) = r. Ær��g S ∈ F r×m � r(S) = r, �$ SM = Ir;
(3’) syybngS���Jyybn�6!8n�
(4) � N ∈ F r×n, r(N) = r. Ær��g T ∈ F n×r � r(T ) = r, �$ NT = Ir;
(4’) JyybngS℄��syybn�6!8n�
(5) 9��% S, T "85R��{
j?5R�
� 13 (1) � A ∈ Fm×n. h rank(A) = r ⇐⇒ �g B ∈ Fm×r, C ∈ F r×m �
rank(B) = rank(C) = r, �$ A = BC.
(2) 9 (1) {% (B,C) "8N�5R�R��C�g B1 ∈ Fm×r, C1 ∈ F r×n
� rank(B1) = rank(C1) = r, �$ A = B1C1, "8�gg�n P ∈ F r×r, �$
B1 = BP , C1 = P
−1C?
(3) � A ∈ Fm×n. h rank(A) = r ⇐⇒ �g r 9CKTU?"5R%�R���gvR9\B7CKhU W1
`y%CKY� ψ1 : V → W1, !%CKY� σ1 : W1 → U , �$ ϕ = σ1ψ1, h�
gCKhU2> θ : W →W1, �$ θϕ = ϕ1, σ1θ = σ.
(1) &��� (��) w W = Imϕ, ψ : V → W , α 7→ ϕ(α), σ "��Y��hy
�Q�
&��� (��) � V " n 7CKhU� ξ1, ξ2, · · · , ξn " V %R9N�� U "
m 7CKhU� η1, η2, · · · , ηm " U %R9N�w
ϕ(ξ1, ξ2, · · · , ξn) = (η1, η2, · · · , ηm)A,
�{ A 6 m × n bn�� r(A) = r. h A = CB, �{ C 6 m × r bn� B 6
r × n bn�� r(B) = r(C) = r. w W " r 7CKhU� ζ1, ζ2, · · · , ζr " W %
R9N� ψ : V →W , σ : W → U "CKY�y�
ψ(ξ1, ξ2, · · · , ξn) = (ζ1, ζ2, · · · , ζr)B,
σ(ζ1, ζ2, · · · , ζr) = (η1, η2, · · · , ηm)C.
hW6 A = CB, \ ϕ = σψ. W6 B Jyy�*S ψ "y��W6 C syy�*
S σ "!��
(2) &��� (��) W6 ϕ = σψ = σ1ψ1 F σ, σ1 !��s Kerψ = Kerϕ =
Kerψ1. � αr+1, αr+2, · · · , αn " Kerψ %N�k6 V %N α1, · · ·, αr, αr+1, · · ·, αn.
h ψ(α1), ψ(α2), · · ·, ψ(αr)" Imψ %N�R" W %N�ψ1(α1), ψ1(α2), · · · , ψ1(αr)
" Imψ1 %N�R" W1 %N�w θ : W → W1,
∑r
i=1 aiψ(αi) 7→
∑r
i=1 aiψ1(αi), h
θ "FqhU%2>�u[Or$ θϕ = ϕ1, σ1θ = σ.
&��� (��) � V " n 7CKhU� ξ1, ξ2, · · · , ξn " V %R9N�� U "
m7CKhU�η1 , η2, · · · , ηm " U %R9N�� W " r 7CKhU�ζ1, ζ2, · · · , ζr
" W %R9N�� U " s 7CKhU� ζ ′1, η
′
2, · · · , η′s " W1 %R9N�w
ϕ(ξ1, ξ2, · · · , ξn) = (η1, η2, · · · , ηm)A,
8
ψ(ξ1, ξ2, · · · , ξn) = (ζ1, ζ2, · · · , ζr)B,
σ(ζ1, ζ2, · · · , ζr) = (η1, η2, · · · , ηm)C,
ψ1(ξ1, ξ2, · · · , ξn) = (ζ ′1, ζ ′2, · · · , ζ ′s)H,
σ1(ζ
′
1, ζ
′
2, · · · , ζ ′s) = (η1, η2, · · · , ηm)G,
�{ A 6 m× n bn� B 6 r× n bn� C 6 m× r bn� H 6 s× n bn�
G 6 m× s bn�<
2>.X�\ A = CB = GH . W6 ψ, ψ1 6y�� σ, σ1
6!��*S r(C) = r(B) = r, r(G) = r(H) = s. mP��g r×m bn C1 �$
C1C = Ir, �g s×m bn G1 �$ G1G = Is, �g n× r bn B1 �$ BB1 = Ir.
[_
s = r(G) ≥ r(C1GH) = r(C1CB) = r(B) = r
= r(B) ≥ r(G1CB) = r(G1GH) = r(H) = s,
\ r = s. w F = G1C, h F " r \5n��� H = FB. ^W6 r(F ) ≥ r(FB) =
r(H) = r, *S F " r \g�n�vR5�� CB = GH = GFB, ℄� B1, $#
GF = C. w θ : W → W1 "CKY�y�
θ(ζ1, ζ2, · · · , ζr) = (ζ ′1, ζ ′2, · · · , ζ ′r)F,
h θ "g�Y�� θϕ = ϕ1, σ1θ = σ.
� 18 � ϕ" n7CKhU V %CK�L�Ær�V = V1⊕V2, �{ V1 ∼= Imϕ,
V2 = Kerϕ.
� 19 � ϕ " n 7CKhU V %CK�L�Ær� ϕ = ψθ, �{ ψ, θ "CK
�L� ψ2 = ψ � θ "g�CK�L�
� 20 � n7CKhU V �CK�L ϕ%OG0E�6 mϕ = (λ−λ1)n1 · · · (λ−
λk)
nk , �{ λ1, · · · , λk ppJV�w Wi = Ker(λ1idV −ϕ)i, 1 ≤ i ≤ k, �{ idV Æ
V �%H&�L�r��
(1) Wi j Wi+1, . i = 1, 2, · · ·;
(2) " i < n1 �� Wi 6=Wi+1;
(3) " i ≥ n1 �� Wi = Wi+1.
&��� oZ ϕ % Jordan
}H�7= λ1 ,ov.X% Jordan i�
9
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