A Q R Q is orthonormal R is upper triangular To find QR decomposition 1 Q Use GramSchmidt to find orthonormal basis for column space of A 2 Let R Q T A Find the QR decomposition of ID: 531976
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Slide1
QR decomposition:
A =
Q
R
Q is orthonormal
R is upper triangular
To find QR decomposition:
1.) Q: Use Gram-Schmidt to find orthonormal basis for column space of A
2.) Let R = Q
T
A
Slide2
Find the QR decomposition of
A =
4
3
1.) Use Gram-Schmidt to find orthonormal basis for column space of ASlide3
Find the QR decomposition of
A =
4
3
1.) Use Gram-Schmidt to find orthonormal basis for column space of A
col(A) = span ,
4
3
{
}Slide4
Find the QR decomposition of
A =
4
3
1.) Use Gram-Schmidt to find
orthogonal
basis for column space of A
col(A) = span ,
4
3
{
}Slide5
Find the QR decomposition of
A =
4
3
1.) Use Gram-Schmidt to find
orthogonal
basis for column space of A
col(A) = span , = span ,
4
3
{
}
{
?
}Slide6
4
3
Slide7
4
3
Find orthogonal projection
of onto
4
3
Slide8
4
3
Find orthogonal projection
of onto
4
3
(1, 2)
(4,
3
)
(1, 2)
(1, 2)
4
3
proj
=
1
(4) + 2(3) 4 + 6
1
2
+ 2
2
1 + 4
= =
Slide9
4
3
Find orthogonal projection
of onto
4
3
(1, 2)
(4,
3
)
(1, 2)
(1, 2)
4
3
proj
=
1
(4) + 2(3)
10
1
2
+ 2
2
5
= =
Slide10
4
3
2
4
Find orthogonal projection
of onto
4
3
(1, 2)
(4,
3
)
(1, 2)
(1, 2)
4
3
2
4
proj
=
1
(4) + 2(3)
1
2
+ 2
2
= =
2
=
Slide11
4
3
2
4
Find orthogonal projection
of onto
4
3
(1, 2)
(4,
3
)
(1, 2)
(1, 2)
4
3
2
4
proj
= =Slide12
4
3
2
4
Find the component
of orthogonal to
4
3
Slide13
2
-1
4
3
2
4
Find the component
of orthogonal to
4
3
− =
4
3
2
4
2
-1 Slide14
4
3
Short-cut for R
2
case:Slide15
Find the QR decomposition of
A =
4
3
1.) Use Gram-Schmidt to find
orthogonal
basis for column space of A
col(A) = span , = span ,
4
3
{
}
{
}
2
-1 Slide16
2
-1
= √ 1
2
+ 2
2
= √
5
= √ (-1)
2
+ 2
2
= √
5
Find the length of each vector:Slide17
Divide each vector by its length:
col(A) = span , =
span ,
= span ,
4
3
{
}
{
}
2
-1
{
}Slide18
{
}
col(A) =
span ,
Q =
A = QR
Slide19
A = QR
A = QR
Q
-1A = Q-1QR
Q-1A = RQ has orthonormal columns:
Thus Q
-1 = QTThus R =
Q-1A = QTASlide20
Find the QR decomposition of
A =
= QR
R = Q-1A = Q
TA = = =
4
3
4
3 Slide21Slide22Slide23Slide24