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DisciplinaÁlgebra Linear II897 materiais7.977 seguidores
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Projeção ortogonal de um vetor v 
sobre o plano definido pelas colunas da matriz A. 
 
 
 
 
 
 
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5
5,0
1
v 
 
 
 
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12
11
12
A 
 
 
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3.6154
2.3462
1.4615
Pv 
 
 
 
 
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0.653850.461540.11538
0.461540.384620.15385-
0.115380.15385-0.96154
AAAAP TT
1
 
 
P = QQ
T
, se Q é obtida da decomposição A = QR. 
 
P é idempotente (P = P
2
) e simétrica (P = P
T
). 
 
Assim, (I \u2013 P) é ortogonal a P, ou seja, (I \u2013 P)TP = 0. 
 
span(P) + span(I \u2013 P) = IR3. 
-2
0
2
-3-2
-10
12
3
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
Projeçao ortogonal de um vetor no plano
y
v 
Pv 
a1 
a2 
(I-P)v 
 
\uf0b7 Transformando o vetor x no vetor \uf0b1 ||x||2e1. 
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1
x \uf0fa
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0
1
1012ex
 
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3
110
12
xexv 
 
 
\uf0b7 A projeção ortogonal de x sobre H é dada por: 
\uf028 \uf029\uf028 \uf029 v
vv
xv
xx
vv
vv
IxvvvvIPx
T
T
T
T
TT
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\uf02d1. 
\uf0b7 ||x||2e1 é o reflexo de x com relação a H: 
Qxx
vv
vv
Iex
T
T
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. 
\uf0b7 A matriz Q é ortogonal (Q
T
Q = I). 
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31623,094868,0
94868,031623,0
Q 
\uf0b7 Poderíamos ter transformado x em \u2013 ||x||2e1. 
Neste caso, teríamos v = x + ||x||2e1. 
Px 
H 
x 
||x||2e1 
x \u2013 ||x||2e1 = v 
DECOMPOSIÇÃO DE HOUSEHOLDER 
 
 
\uf0b7 Algoritmo da Decomposição de Householder. 
 
1. Para i = 1 até n, 
1.1. x = Ai:m,i; 
1.2. vi = x + sign(x1)||x||2e1; 
1.3. vi = vi / ||vi||2; 
1.4. Ai:m,i:n = (I \u2013 2vivi
T
)Ai:m,i:n; 
 
Custo computacional: 32
3
2
2 nmn \uf02d
 operações. 
 
 
\uf0b7 Resolução do sistema Ax = b: 
 
 
Se A = QR, temos QRx = b, ou seja, Rx = Q
T
b. 
 
Fazendo Qi = (I \u2013 2vivi
T
), temos Q
T
 = QnQn-1 ... Q1. 
 
 
\uf0b7 Algoritmo da resolução do sistema Ax = b: 
 
1. y = b; 
2. Para i = 1 até n, 
2.1. yi:m = Qi yi:m; 
3. Resolver o sistema Rx = y; 
 
 
\uf0b7 Decomposição de Householder de 
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10
12
21
A : 
 
1
o
. passo: 
 
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0
2
1
x , 
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0
0.52573
0.85065
v1 , 
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100
00.447210.89443-
00.89443-0.44721-
Q1 
 
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1-0
1.3416-0
1.7889-2.2361-
AQA 11 
 
2
o
. passo: 
 
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1-
1.3416-
x , 
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0.31481-
0.94915-
0
v2 
 
(v2 já está normalizado e tem primeiro elemento nulo) 
 
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0.801780.59761-0
0.59761-0.80178-0
001
Q2 
 
R
00
1.67330
1.7889-2.2361-
AQQAQA \uf03d
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\uf03d\uf03d\uf03d 12122 
\uf0b7 Resolução de um sistema Ax = b. 
 
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10
12
21
A , 
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1
1
1
b 
 
(observe que 
)(ARb\uf0ce
) 
 
 
Calculando y = Q
T
b = Q2Q1b. 
 
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1
1.3416
0.44721-
bQy 1 , 
 
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0
1.6733-
0.44721-
yQy 2 . 
 
(como y3 = 0, o sistema é compatível) 
 
 
Resolvendo o sistema Rx = y. 
 
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0
1.6733-
0.44721-
x
x
00
1.67330
1.7889-2.2361-
yRx
2
1
 
 
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1
x