Universidad del Bío-Bío Facultad de Educación y Humanidades Escuela de Pedagogía en Educación Matemática
F ACTOR ACTORI ZACI ÓN L U
Profesor Guía:
Dr. Luis Alberto Friz Roa Autores:
Marcos Alejandro Torres Solís Natalia Angélica Villalobos Villalobos Castillo Castillo SEMINARIO PARA OPTAR AL TÍTULO DE PROFESOR EN EDUCACIÓN MATEMÁTICA Chillán, 2015
´ FACTORIZACI ON LU
Autores : Marcos T orres, orres, N atali atalia a V illalo illalobo boss
P rofeso rofesorr Gu´ıa ıa : Dr. Dr. Luis uis F riz riz Roa Roa
2015
´ FACTORIZACI ON LU
Autores : Marcos T orres, orres, N atali atalia a V illalo illalobo boss
P rofeso rofesorr Gu´ıa ıa : Dr. Dr. Luis uis F riz riz Roa Roa
2015
LU
P A = LU
m n n p
Ax = B n
n n
LU LDU LU
A=
A
LU
U
L
LU
LU A = LU
A L
U LU
A ∈ M nxn(R) L, U ∈ M nxn (R) A = LU Ax = b LUx = b ←→ [ Lz = b ∧ U x = z ] LU
=
1 l21
0 1
ln−1,1 ln−1,2 ln1
ln2
a11
a12
a21
a22
··· ···
··· ···
an2
0 0
a1n
a2,n−1
a2n
··· · · · an−1,n−1 an−1,n ann · · · an,n−1
an−1,1 an−1,2 an1
a1,n−1
0 0 0
··· ··· 1 · · · ln,n−1 1
u11 u12 · · ·
u1,n−1
u1n
u22 · · ·
u2,n−1
u2n
0 0 0
··· · · · un−1,n−1 un−1,n 0 unn ···
0 0
A A
A
n
A
LU L A = aij
n n
A
n
| a ij |>
aij paratodafilai = 1,...,n
j =1 ,j =i
n LU
LDU
LU LU
A m
m n
m n
n
A
=
a1 j
a22
··· ···
ai2
···
a11
a12
a21 ai1
am1 am2 · · ·
A = (aij )
B = (bij )
a1n
a2 j
··· ···
aij
···
a1n
amj · · ·
amn
a2n
A = (aij )
B = (bij )
m n
A
m n A + B
B
A + B = (aij + bij
)=
A + B A
··· ···
a1n + b1n
am1 + bm1 am2 + bm2 · · ·
amn + bmn
a11 + b11
a12 + b12
a21 + b21
a22 + b22
a2n + b2n
m n
B
A = (aij )
m n
α
m n αA,
αA = (αaij
)=
··· ···
αa1n
αam1 αam2 · · ·
αamn
m n
α
αa11
αa12
αa21
αa22
A,B,C
A + 0 = A ,0 ∈ M mxn(R)
0A = 0
0 ∈
R
A + B = B + A
(A + B ) + C = A + (B + C ) α(A + B ) = αA + αB
αa2n
β
1A = A (α + β )A = αA + βA M nxn(R)
A = (aij )
B = (bij )
m n A
n p
m p C = (cij )
B
cij = (filaideA) · (columna j de B ) ij j
AB
i
B. n
cij = a i1 b1 j + ai2b2 j + · · · + ain bnj
=
aiq bqj
q=1
A A
B
B
I n
n n
I n = (bij ) donde
n n
bij
1 = 0
si
i = j,
si
i = j.
A
A
n n AI n = I n A
I n
n n I m
A
B
n n AB = BA = I
B
A−1
A AA−1 = A −1 A = I
A
A
A = (aij )
m n
At
A
n m At = (a ji )
A.
A
=
··· ···
a1n
am1 am2 · · ·
amn
a11
a12
a21
a22
a2n
At
=
··· ···
an1
a1m a2m · · ·
anm
a11
a21
a12
a22
an2
A = (aij )
(bij )
n m
B=
m p
(At )t = A (AB )t = B t At A
B
(A + B )t = A t + B t
n m At
A
A
At = A
n n
A
A
P
A A.
A ∈ M mxn (R) A
A
A
(At )−1 = (A−1 )t
A ∈ M nxn (R) A i
j
e1 [i → j ]
=0 λ
i e2 [i → λi ] i A
= 0 (e3 [i → i + λj ] λ
j A,B ∈ M nxn(R)
B
A ∼ B, B
A
A ∈ M nxn (R) I n
A
m
A
I m
A
e1 [i → j ] e2 [i → λi ] e3 [i → i + λj ] Anxn
A
=
a11 a12 .... a1n a21 a22 .... a2n
an1 an2 .... ann
I n
n n
e1 j n n
i P A = B
B
=i l
j l =
q l =
plq = 0 i
pll = 1
P
I
j n
blk
=
plq aqk = a lk
q=1
A
i
j
P
= j q
piq = 0
i
pij = 1
n
bik
=
piq aqk = a jk
q =1
i
j
P j
A ∈ M nxn (R)
e2 P
n n
i
α. P A = B
= i l
B
= q l
plq = 0
n n pll = 1
n
blq
=
plq aqk = a lq
q=1
i piq = 0
i n
biq
= q =1
piq aqi
= q i
pii = α
pii = α
= i k
pik = 0
n
pik akq = αa iq
k=1
biq = αa iq
e3
i
α
P j
n n α
P A = B
0
,
pik
i, k = j k=
α,
k = j
1,
k = i n
bik
= q =1
m n
j
piq aqk = αa jk + aik
i
A
B
n n AB = BA = I
B
A−1
A AA−1 = A −1 A = I
A
A n 2n
A
A I n
I n
A A
[C D]
I n C = I n
D = A −1
I n C =
C
A
A−1
[A I ] · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · I A
−1
A I m
(E n . . . . . . E3 E 2 E 1)A = I E 1E 2 E 3.....E n A E n . . . . . . E3 E 2 E 1 = A −1 ,
A
1 = 1
1 0 0 1 0 1 0
luego A−1 A = I
A−1
A
1 1
I
1 0
1 0
0 1
0 1
0 1 0
1
2
→
2
0 0
0 0 1
1
1 0 0
1
0
−1 1 1
0
1
0
−1 1 0
0 0
0 1
2
3
→
3
2
1 0 0
3
2
→
2
0
1
→
1
0
=
2
→
−1 1 0
1
1
0
−1 1
−1 1 1
1
0
1
0
−1 0
0
0
0
1
−1 1
0 −1 1
2
1 0 5
0
3
1 0 4
1
0 0
0
0
1
0
−1 0
0
0
0
1
−1 1
−1 −1 1
2
1 0
0 0
1
0
1 0
0
0
0 0 1
−1 1
−1 1 1
A
A−1
=
1 0 −1 0 0 1 −1 1 1
E 5 E 4 E 3 E 2 E 1 = A −1
1 0
0 1 0 0
0 −1 0 0 1
0 1 0 0
1 1 0 0 1
0
0 0 1 −1 0 1 1 0 0 1 0 1 0 −1 0 1 1
0
0 0 1 0 0 1
=
A −1
LU LU
A, n n A = LU L
n n n Ax = b A = LU
Ax = b
L(U x) = b y = U x
n
U
LU
Ly = b y
U x = y
LU
A
n n.
A U L
A = LU A
E 1, E 2 ,...,E k
E k E k−1 · · · E 2 E 1 = U A = E 1−1 E 2−1 · · · E k−1U.
U
LU E 1, E 2 ,...,E k E 1−1 , E 2−1 , · · · , E k−1
L= E 1−1 E 2−1 · · · E k−1 LU
A A = LU A LU
A = L 1 U 1 = L 2 U 2 , U 1 , U 2
1 L− L1 = U 2 U 1−1 2
1 L− L1 = I 2
L1 = L 2 U 1 = U 2
A
LU
A
=
2 1 1 4 1 0 −2 2 1
1 = −2 u v
7
w
F 2 → F 2 − 2F 1
2 1 1 0 −1 −2 −2 2 1
F 3 → F 3 + F 1
2 0
1 −2
2 0
1 −1 0 3
2
F 3 → F 3 + 3 F 2
1 1 −1 −2 0 0 −4
A U
Ux
2 = 0
1 1 −1 −2 0 0 −4
1 = −4 u v
w
−4
LU
”triangular superior” A E E 1
E 1
1 = −2 0
E 2
1 = 0
0 0 1 0 1 0 1
E 1 , E 2
0 0
0 1 0 1
y
E 3
1 = 0
0 0 1 0 0 3 1
E 3
I A E 3 E 2 E 1 A = U
A
U
E 3E 2 E 1
1 = −2
0 0 1 0 −5 3 1
U
U
LU
A
U U
A sumamos E 1
E 1−1
E 1
1 = 2
0 0 1 0 0 0 1
E 1−1
E 1−1
E 1
E 1 E 1−1 = I ,
E 1−1 E 1 = I . E 2
−1
E 2
=
0 0
1 0 0 1 −1 0 1
E 3
−1
,
E 3
1 = 0
0 0
0 1 0 −3 1
U A U
A
U
A
A U A = E 1−1 E 2−1 E 3−1 U
LU L U
A E 1−1 E 2−1 E 3−1 = L,
L
1 = 2
0 0
0 1 0 0 1
de este modo A = LU
0 1 0 0
1 0 0 1 −1 0 1
0 0 =
0 1 0 −3 1
0 0
1 0 2 1 −1 −3 1
L U L
E 3E 2 E 1L
1 = −2
0 0 1 0 −5 3 1
1 0 0 2 1 0 −1 −3 1
1 = 0
0 0 1 0 0 0 1
E 3E 2 E 1 = L −1 A = LU
60x1 + 30x2 + 20x3 = 180 30x1 + 20x2 + 15x3 = 115 20x1 + 15x2 + 12 x3 = 86
LU Ax = b
60 30
30 20 20 15 20 15 12
180 = 115 x1 x2
86
x3
U 1
2
→
2
−
1 2
1
60 0
20 5
30 5 20 15 12
2
3
→
3
−
1 3
1
60 0
30 20 5 5 5 16 3
60 0
30 20 5 5 0 13
0
3
3
→
3
−
2
0
U
60 = 0 0
30 20 5 5 0 13
LU
L
1 =
0 0 1 0 1 1
1 2 1 3
A
60 30
30 20 20 15 20 15 12
1 = 1 2 1 3
0 0 1 0 1 1
60 0 0
30 20 5 5 0 13
Ly = b
1 1 2 1 3
y1 1 2 1 3
y1 +
y2
y1+ y2 + y3
0 180 0 = 115
0 1 1 1
y1 y2
86
y3
= 180 = 115 = 86
y1 = 180
y
y2 = 25
y3=1
180 = 25 1
U x = y
LU
60 0 0
30 20 5 5 0 13
180 = 25 x1 x2
1
x3
60x1 + 30x2 + 20x3 = 180 5x2 + 5x3 = 25 1 = 1 x 3 3
x3 = 3
x2 = 2
x
x1 = 1
1 = 2 3
LU
LU
P A = LU LU P A = LU
P LU
LU P A = LU
P A = LU
A ∈ M nxn (R)
P L U
n n n n (lii = 1) n n
P A = LU
LU
P A = LU
A
0 = −1
0 2 5 −2 6 7
3
0 −1 3
F 2
,
3 6 7 ←→ −1 5 −2 0 0 2 3 6 7 1 −→ + 0 7 3 0 0 2 3 6 7 = 0 7
0 2 5 −2 6 7
F 1
F 3
F 1
1
F 2
3
1
U
3
0 0 3
L
L
1 = −
1 3
0
0 0 1 0 0 1
P F 1 ←→ F 3
I
1 = 0
0 0 1 0 0 0 1
F 1 ←→ F 3
0 0
0 1 1 0 1 0 0
LU
P
0 0
1 0 0 −1
0 1 1 0 0
3
0 5 6
0 = 0
0 1 1 0 1 0 0
2 1 −2 = −
1 3
7
0
P A = LU
0 3 0 0
0 1 0 1
7
6 7 13 0 0 2
LU U
L U
D
d1, d2 ,...,dn A = LDU
A L
U D U A = LU
A
=
=
2 1 1 0 −1 −2 0 0 −4 2 0 0 1 0 0 0 −1 0 0 1 2 =
1 0 2 1 −1 −3 1
1 0 0 2 1 0 −1 −3 1
0
0
−4
1
1
2
2
0 0 1
D
LDU
U
U
A ∈ Mnxn (R)
P A = LU
L ∈ Mnxn (R)
A
LU
P ∈ Mnxn (R)
d1, d2 ,...,dn
U
d1 , d2 ,...,dn A = LDU
D A
L L
D U
A
PA
U
P
LU
U
0 di = i = 1, 2,...,n)
U
i = 1, 2,...,n
U U = DU
di
P A = LU = LDU
A A A = L 1 D1 U 1
A = L 2 D2 U 2
L
U D L1 = L2 D1 = D 2 U 1 = U 2 A
LDU
L1D1 U 1 = L 2 D2 U 2 1 L− 1
L1 U 2−1
U 2U 2−1 = I D1
d1 d2 . . . dn
−1
=
1 L− 1
1
d1 1
d2
. . . 1
d
n
D1−1
U 2−1 1 U 1 U 2−1 = D 1−1 L− L2 D2 1
U 1 U 2−1 = I U 1 = U 2
U 2 L1 = L 2
D1 = D 2
LDU
A
E 12 (−2)
E 13 (1)
1 −1 −1 2 0 −3 −1 7 −1
E 23 (−3)
¯ A
A
=
−1 −3
1 −1 2 0 −1 7 −1
1 −1 −1 0 2 −1 −1 7 −1
1 0
1 0
−1 −1
−1 −1 2 −1 0 6 −2
−1 2 0 0
1
A A
L
,
L = [E 23 (−3)E 13 (1)E 12 (−2)]
−1
= E 12(−2)−1 E 13(1)−1 E 23(−3)−1
= E 12(2)E 13(−1)E 23(3)
=
0 0
1 0 2 1 −1 3 1
LU
U
A
=
1 = 0
−1 2 0 0
1 0 0 2 1 0 −1 3 1
LU
A
−1 −1 1
1 0
−1 −1 2 −1 0 0 1
A
LDU
A
L
U
D A
D
1 = 0
0 0 2 0 0 0 1
D
D
U DU
A
U
U
1 = 0
−1 −1 1 − 12 0 0 1
LDU
A
=
0 1 0 0
1 0 2 1 −1 3 1
A
0 1 0 0
0 2 0 0 1
−1 1 0 0
−1 − 1 2
1
