ejemplo de factorización plu - egor...
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Ejemplo de factorizacion PLU
Egor Maximenkohttp://www.egormaximenko.com
Instituto Politecnico Nacional,Escuela Superior de Fısica y Matematicas
Mexico, D.F.
5 de enero de 2015
PrerrequisitosAlgoritmo de factorizacion LU
(guardando L y U juntas)Matrices de permutacion
Factorizacion PLU
Problema de factorizacion PLU
Hallar una factorizacion PLU de la matriz
A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
.
Se buscan matrices P, L, U tales quePA = LU,P es una matriz de permutacion,L es unitriangular inferior,U es triangular superior.
Una matriz puede tener varias factorizaciones PLU.En este ejemplo una respuesta posible es
P =
1 0 0 00 0 1 00 0 0 10 1 0 0
, L =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
, U =
5 2 4 30 3 −1 30 0 1 00 0 0 −2
.
Problema de factorizacion PLU
Hallar una factorizacion PLU de la matriz
A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
.
Se buscan matrices P, L, U tales quePA = LU,P es una matriz de permutacion,L es unitriangular inferior,U es triangular superior.
Una matriz puede tener varias factorizaciones PLU.En este ejemplo una respuesta posible es
P =
1 0 0 00 0 1 00 0 0 10 1 0 0
, L =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
, U =
5 2 4 30 3 −1 30 0 1 00 0 0 −2
.
Problema de factorizacion PLU
Hallar una factorizacion PLU de la matriz
A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
.
Se buscan matrices P, L, U tales quePA = LU,P es una matriz de permutacion,L es unitriangular inferior,U es triangular superior.
Una matriz puede tener varias factorizaciones PLU.
En este ejemplo una respuesta posible es
P =
1 0 0 00 0 1 00 0 0 10 1 0 0
, L =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
, U =
5 2 4 30 3 −1 30 0 1 00 0 0 −2
.
Problema de factorizacion PLU
Hallar una factorizacion PLU de la matriz
A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
.
Se buscan matrices P, L, U tales quePA = LU,P es una matriz de permutacion,L es unitriangular inferior,U es triangular superior.
Una matriz puede tener varias factorizaciones PLU.En este ejemplo una respuesta posible es
P =
1 0 0 00 0 1 00 0 0 10 1 0 0
, L =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
, U =
5 2 4 30 3 −1 30 0 1 00 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1
=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1
=========
5 2 4 3
3 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1
R3 +=−6R1R4 += 5R1
=========
5 2 4 33
0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1
R4 += 5R1
=========
5 2 4 33
0 0 −2
6
3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33
0 0 −2
6
3 −1 −3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0
0 −2
6
3 −1 −3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0
−2
6
3 −1 −3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26
3 −1 −3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3
−1 −3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1
−3
−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5
−12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12
5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5
12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2
=========
5 2 4 36 3 −1 −33
0 0 −2
−5
−4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2
R4 += 4R2
=========
5 2 4 36 3 −1 −33 0
0 −2
−5
−4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0
0 −2
−5 −4
1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0
−2
−5 −4
1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4
1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1
0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Solucion (en cada paso PA = LU)
P1,2,3,4A =
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
R2 +=−3R1R3 +=−6R1R4 += 5R1=========
5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12
R2↔R3−−−−→
P1,3,2,4 A =
5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12
R3 += 0R2R4 += 4R2=========
5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0
R3↔R4−−−−→
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
R4 += 0R3=========
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
El ultimo estado de la solucion es
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Respuesta:1 0 0 00 0 1 00 0 0 10 1 0 0
︸ ︷︷ ︸
P
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
︸ ︷︷ ︸
A
=
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
︸ ︷︷ ︸
L
5 2 4 30 3 −1 30 0 1 00 0 0 −2
︸ ︷︷ ︸
U
.
El ultimo estado de la solucion es
P1,3,4,2 A =
5 2 4 36 3 −1 −3−5 −4 1 0
3 0 0 −2
.
Respuesta:1 0 0 00 0 1 00 0 0 10 1 0 0
︸ ︷︷ ︸
P
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
︸ ︷︷ ︸
A
=
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
︸ ︷︷ ︸
L
5 2 4 30 3 −1 30 0 1 00 0 0 −2
︸ ︷︷ ︸
U
.
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5
2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2
4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4
330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30
12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3
24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1
18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3
−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25
− 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12
− 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1
− 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 0
15 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015
6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0
12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0
9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 330 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15
−25 −22 −15 −315 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15
−25 −22 −15 −315 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
Comprobacion
LU =
1 0 0 06 1 0 0−5 −4 1 0
3 0 0 1
5 2 4 30 3 −1 30 0 1 00 0 0 −2
=
5 2 4 3
30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
.
P1,3,4,2 A = P1,3,4,2
5 2 4 3
15 6 12 730 15 23 15−25 −22 −15 −3
=
5 2 4 3
30 15 23 15−25 −22 −15 −3
15 6 12 7
. X
EjercicioConstruir una factorizacion PLU de la siguiente matriz y hacer la comprobacion.
A =
0 −2 1 −23 1 5 66 0 11 66 8 1 20
.
¡Ejercer para aprender!
Una respuesta posible:
P =
0 1 0 01 0 0 00 0 0 10 0 1 0
, L =
1 0 0 00 1 0 02 −3 1 02 1 0 1
, U =
3 1 5 60 −2 1 −20 0 −6 20 0 0 −4
.
EjercicioConstruir una factorizacion PLU de la siguiente matriz y hacer la comprobacion.
A =
0 −2 1 −23 1 5 66 0 11 66 8 1 20
.
¡Ejercer para aprender!
Una respuesta posible:
P =
0 1 0 01 0 0 00 0 0 10 0 1 0
, L =
1 0 0 00 1 0 02 −3 1 02 1 0 1
, U =
3 1 5 60 −2 1 −20 0 −6 20 0 0 −4
.
EjercicioConstruir una factorizacion PLU de la siguiente matriz y hacer la comprobacion.
A =
0 −2 1 −23 1 5 66 0 11 66 8 1 20
.
¡Ejercer para aprender!
Una respuesta posible:
P =
0 1 0 01 0 0 00 0 0 10 0 1 0
, L =
1 0 0 00 1 0 02 −3 1 02 1 0 1
, U =
3 1 5 60 −2 1 −20 0 −6 20 0 0 −4
.