Math 221: LINEAR ALGEBRA 8-4. QR Factorization Le Chen 1 Emory - - PowerPoint PPT Presentation

Math 221: LINEAR ALGEBRA

§8-4. QR Factorization

Le Chen1

Emory University, 2020 Fall

(last updated on 08/27/2020) Creative Commons License (CC BY-NC-SA) 1Slides are adapted from those by Karen Seyffarth from University of Calgary.

The QR Factorization

Definition

Let A be a real m × n matrix. Then a QR factorization of A can be written as A = QR where Q is an orthogonal matrix and R is an upper triangular matrix.

The QR Factorization

Definition

Let A be a real m × n matrix. Then a QR factorization of A can be written as A = QR where Q is an orthogonal matrix and R is an upper triangular matrix.

Theorem

Let A be a real m × n matrix with linearly independent columns. Then A can be written A = QR with Q orthogonal and R upper triangular with non-negative entries on the main diagonal.

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization. Apply the Gram-Schmidt Process to the columns of . Label the resulting columns respectively. Defjne . Let be constructed as . Then is

• rthogonal.

Let be constructed as . . . . . . . . . . . . The result is .

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization.

• 1. Apply the Gram-Schmidt Process to the columns of A. Label the

resulting columns B1, B2, · · · , Bn respectively. Defjne . Let be constructed as . Then is

• rthogonal.

Let be constructed as . . . . . . . . . . . . The result is .

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization.

• 1. Apply the Gram-Schmidt Process to the columns of A. Label the

resulting columns B1, B2, · · · , Bn respectively.

• 2. Defjne Ci =

1 BiBi.

Let be constructed as . Then is

• rthogonal.

Let be constructed as . . . . . . . . . . . . The result is .

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization.

• 1. Apply the Gram-Schmidt Process to the columns of A. Label the

resulting columns B1, B2, · · · , Bn respectively.

• 2. Defjne Ci =

1 BiBi.

• 3. Let Q be constructed as Q =
• C1

C2 · · · Cn

• . Then Q is
• rthogonal.

Let be constructed as . . . . . . . . . . . . The result is .

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization.

• 1. Apply the Gram-Schmidt Process to the columns of A. Label the

resulting columns B1, B2, · · · , Bn respectively.

• 2. Defjne Ci =

1 BiBi.

• 3. Let Q be constructed as Q =
• C1

C2 · · · Cn

• . Then Q is
• rthogonal.
• 4. Let R be constructed as

R =        B1 A2 · C1 A3 · C1 · · · An · C1 B2 A3 · C2 · · · An · C2 B3 · · · An · C1 . . . . . . . . . . . . · · · Bn        The result is .

Finding the QR Factorization

Let A be a real m × n matrix with linearly independent columns A1, A2, · · · , An. The following procedure results in the QR factorization.

• 1. Apply the Gram-Schmidt Process to the columns of A. Label the

resulting columns B1, B2, · · · , Bn respectively.

• 2. Defjne Ci =

1 BiBi.

• 3. Let Q be constructed as Q =
• C1

C2 · · · Cn

• . Then Q is
• rthogonal.
• 4. Let R be constructed as

R =        B1 A2 · C1 A3 · C1 · · · An · C1 B2 A3 · C2 · · · An · C2 B3 · · · An · C1 . . . . . . . . . . . . · · · Bn       

• 5. The result is A = QR.

Example

Let A =   4 1 2 3 1   Find the QR factorization of A.

Example

Let A =   4 1 2 3 1   Find the QR factorization of A.

Solution

First we apply the Gram-Schmidt Process to the columns of A. B1 = A1 =   4 2   B2 = A2 − A2 · B1 B12 B1 =   1 3 1   − 10 20   4 2   =   −1 2 1  

Solution (continued)

Next, normalize these vectors. C1 = 1 B1B1 = 1 √ 20   4 2   C2 = 1 B2B2 = 1 √ 6   −1 2 1  

Solution (continued)

Next, normalize these vectors. C1 = 1 B1B1 = 1 √ 20   4 2   C2 = 1 B2B2 = 1 √ 6   −1 2 1   Then the matrix Q is constructed using these vectors as columns. Q =   

4 √ 20

− 1

√ 6 2 √ 20 2 √ 6 1 √ 6

   =   

2 √ 5

− 1

√ 6 1 √ 5 2 √ 6 1 √ 6

  

Solution (continued)

Now construct R: R = B1 A2 · C1 B2

• =

√ 20 √ 5 √ 6

Solution (continued)

Now construct R: R = B1 A2 · C1 B2

• =

√ 20 √ 5 √ 6

• Therefore

A = QR   4 1 2 3 1   =   

2 √ 5

− 1

√ 6 1 √ 5 2 √ 6 1 √ 6

   √ 20 √ 5 √ 6