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QR decomposition: QR decomposition:

QR decomposition: - PowerPoint Presentation

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QR decomposition: - PPT Presentation

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

orthogonal find decomposition span find orthogonal span decomposition space col gram schmidt basis column orthonormal projection proj vector component length

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Presentation Transcript

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 Slide21
Slide22
Slide23
Slide24