第七讲：矩阵的秩

A ∈ F m×n, B ∈ F n×t.
iz §k© d eV 34 xS 1
rank(A) = r,
P, Q,
A=P
Er 0 00
Q.
rank(AB) =
rank(
Er 0 00
QB) ≥ rank(QB) − (n − r) = rank(A) + rank(B) − n.
iz §k©d eV 34 2
1. rank(A) = rank(A ).
2. rank(kA) =
rank(A) k = 0
0
k=0
3. rank(A) = rank(A).
h n
h 4. rankA = 1 h 0
rank(A) = n rank(A) = n − 1 rank(A) < n − 1
d u 4e 5. P
A ∈ Rm×n A ∈ F n×m.
rank(A A) = rank(A).
1
d D sepV txv sp |hiF rank(A) = r.
A
l1 · · · lr j1 · · · jr
= 0,
s
(s > r)
C = (A A) r
C
j1 · · · jr j1 · · · jr
=
A
t xS F m×s, Q2 ∈ F s×m,
A = P1Q1, B = P2Q2.
A + B = P1 P2
Q1 Q2
.
.
rank(A + B) ≤ rank(P1, P2) ≤ rank(A) + rank(B).
iizz §§kk©© 4
rank(A + B) ≤ rank
A A+B 0B
m−s n−t
4 FwEDeQR C ∈ F s×t. rank(A) ≥ r + s + t − m − n.
rank(B) ≥ r + s − m.
(II)
1. rank
A0 0B
= rank(A) + rank(B).
2. rank
A0 CB
≥ rank(A) + rank(B).
iz §k© d h y 3. rank(AB) ≤ min{rank(A), rank(B)}.
|C C| =
C
1≤l1 <···<lr ≤m
1 ··· r l1 · · · lr
C
l1 · · · lr 1 ··· r
=
A
l1 · · · lr 1 ··· r
A
l1 · · · lr 1 ··· r
> 0.
1≤l1<···<lr ≤m
xS rank(A A) = rank(C C) = r.
1
rank(A) = r. s > r
AB
i1 · · · is j1 · · · js
=
A
1≤l1 <···<ls ≤m
i1 · · · is l1 · · · ls
B
l1 · · · ls j1 · · · js
= 0.
2
xSiz §k© d C rank(AB) ≤ r = rank(A).
dimImA + dimImB. rank(A + B) ≤ rank(A) + rank(B).
iz §k© d D6&'h D6&'h D6&'h 7
(A + B)X = 0
V , AX = 0
W, BX = 0
U.
e h xS V ⊇ W ∩ U.
dim(W + U ) + dim(W ∩ U ) = dim(W ) + dim(U ),
BX = 0
C rank(B) ≥ rank(AB).
ABX = 0
{BX = 0
BAX=0 AX=0
} ⊆ {ABX = 0
}.
rank(A) ≥ rank(AB).
iz §k© d e 4
A = (A1A2 · · · An), B = (bij )ns.
xS D"# S D"#$%l D n l=1
rank(AB) ≤ rank(B).
2
rank(A) = r.
A=P
Er 0 00
Q, rank(AB) = rank(
Er 0
0 0
QB) ≤ r.
d rank(B) = s. B = P1
Es 0 00
Q1, rank(AB) = rank(AP1
Er 0
0 0
) ≤ s.
iz §k© D6u D6 D6&' D6&' 3
iz §kF 6F 3
A AX = 0 AX = 0
` 7. (1). A
( ), rank(A)
1.
d fd e (2). A ∈ F m×n, rank(A) = r. A
s
B ∈ F s×n,
d fc BX3 (3). A ∈ Fm×n, rank(A) = r. A
blsAl).
AB
A
"# rank(AB) ≤ rank(B).
AB = (
n l=1
bl1
Al
,
n l=1
bl2
Al
,
· · ·,
rank(AB) ≤ rank(A).
AB
iz §k© d eV 34 5 xS Dr 0 .
rank(A) = r, rank(B) = s. rank(AB) = rank((P A)(BQ)) = rank(
rank
A B
=
n−dim(W ∩U ) = (n−dim(W ))+(n−dim(U ))+dim(W +U )−n≤ (n−dim(W ))+(n−dim(U )) =
rank(A) + rank(B).
iz §k© 5
rank(A, B) ≤ rank
AB 0B
≤ rank
A0 0B
= rank(A) + rank(B).
s+1
B.
Laplace
r+1
5. rank(A + B) ≤ rank(A) + rank(B).
iz §k©d u D&'De u D&'DeFe 1
Ai1 , Ai2, · · · , Air A
Bj1 , Bj2, · · · , Bjs B
A+B
D¢9 "#£ S ¤¥$%l rank(A + B) ≤ rank{Ai1, Ai2, · · · , Air, Bj1, Bj2, · · · , Bjs} ≤
s
1
0 0
p
;
yhi
⇐⇒ ⇐⇒
A(())00
AX
=
0
vD6r r&9; 'DDt(7)w()u()D$%0
⇐⇒
AX = 0
n − r;
d $%&'D`9e $%12 fX Dt7w ⇐⇒ n
α1, α2, · · · , αn,
A A, ImA
r;
dv$%12 ⇐⇒
A : F n → F n, X → AX, dimImA = r;
Q,பைடு நூலகம்AQ = (C, 0),
xS a11 · · · a1r
C = · · · · · · · · · rank(C) = r.
QA =
am1 · · · amn
A 0
rank(A A) = rank( A (C, 0) = rank( C C 0 ) = rank(C C).
0
00
iz §k© Y 6
(A, B) = (A, 0) + (0, B). ”rank(A + B) ≤ rank(A) + rank(B)”.
iz §k© d f D}~ sp v 7
rank(A) = r, rank(B) = s. (A, B)
r+s+1
B v B fY C ¡phiF A
rank(AP ) = rank(P A)rank(A).
ij§k© 4 lhvm9n734Dopq r on734hwf34D sXD ( )
()
n712tn712gu13v4wDxEFh yv 6. rank(A A) = rank(A);
iz §kF Y {Fd 1
Binet-Cauchy
P, Q,
Cr 0
Dr
PA =
Cr 0
, BQ =
0 ) = rank(
CD 0 00
)
iz ≤ r, s. §k©
6
Im(AB) ⊆ ImA, KerB ⊆ Ker(AB).
iz §k© d 4. rank(A, B) ≤ rank(A) + rank(B). iz 1 §k© Yra"nk#(A$)%=r, ranrk0(BF)
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d A ⇐⇒ ⇐⇒ ⇐⇒
⇐⇒
⇐⇒
⇐⇒ ⇐⇒ ⇐⇒
AAAAAAA∈ VFfffDDDDm gvv×h``w&&""n3,i''99##e4DDDsDDrErpp0Psarnttrrp∈kuuD(00DFAqgm)rg;r00×;=77msxxwr,wtiQiu""∈##xrFx;99vnv×ttrnhh,+rr1;;DsPpAhQpy=hDirE0+r
=
s.
YDn712
(A,
B)

r
+
s
yhiF
iz 2 §k© 3
A B
D()D9tF
X =0
iz §k© d D6&'h D6&'h D6&'h 4
A B
X =0
V , AX = 0
W, BX = 0
e h xS U. V ⊆ W ∩ U.
dim(W + U ) + dim(W ∩ U ) = dim(W ) + dim(U ),
n − rank(A + B) =
dim(V ) ≥ dim(W ∩ U ) = dim(W ) + dim(U ) − dim(W + U ) ≥ (n − rank(A)) + (n − rank(B)) − n.
¦ rank(A + B) ≤ rank(A) + rank(B).
6. rank(AB) ≥ rank(A) + rank(B) − n,
iz §k©d V 34 r + s = rank(A) + rank(B).
2
rank(A) = r, rank(B) = s.
P1, P2, Q1, Q2,
A = P1CQ1, B =
3
¦ e P2DQ2,
C = rank
Er 0
0 0
, D = rank
Es 0
0 0
.
A+B = (P1, P2)
C0 0D
Q1 Q2
xS rank(A + B) ≤ rank(
C 0
0 D
Q1 Q2
)≤ rank(
C 0
0 D
) = rank(A) + rank(B).
iz §k© d V 3
A, B ∈ F m×n, rank(A) = r, rank(B) = s.
P1 ∈ F m×r, Q1 ∈ F r×m, P2 ∈
1≤l1 <···<lr ≤m
j1 · · · jr l1 · · · lr
A
l1 · · · lr j1 · · · jr
=
A
xS rank(C) ≥ r.
fw D}~ sp C
s
1≤l1 <···<lr ≤m
(s > r)
l1 · · · lr j1 · · · jr
A
l1 · · · lr j1 · · · jr
= rank
A0 0B
= rank(A) + rank(B).
5
rank(A + B) ≤ rank(A + B, B) = rank(A, B) ≤ rank(A) + rank(B).
iz §k© h xS 6
Im(A + B) ⊆Im(A)+Im(B).
dimIm(A + B) ≤ dimImA+ImB) ≤
rank(A) = r, rank(B) = s,
xS B = P1
Es 0 00
Q1.
d ¦ ev QP1 =
C11 C12 C21 C22
rank(AB) = rank(
,
C11 ∈ F r×s.
P, P1 , Q, Q1,
V34 ⇐⇒
P ∈ F m×r, Q ∈ F r×n, rank(P ) = rank(Q) = r,
A = P Q;
V 9$%D 9$%D ⇐⇒
r
α1, α2, · · · , αr ∈ F 1×n, r
β1, β2, · · · , βr ∈ F m×1,
gFwEDQR A = β1α1 + β2α2 + · · · + βrαr. (I)
> 0.
C
i1 · · · is j1 · · · js
=
A
1≤l1 <···<ls ≤m
i1 · · · is l1 · · · ls
A
l1 · · · ls j1 · · · js
= 0,
xS rank(C) ≤ r. rank(C) = r.
iz §k© d eV 4 f 2
A ∈ F m×n, rank(A) = r = rank(A ).