5.8. EXERCISES 139

Theorem 5.7.5 Let A be any real m × n matrix. Then there exists an orthogonal matrixQ and an upper triangular matrix R having nonnegative entries on the main diagonal suchthat

A = QR

and this factorization can be accomplished in a systematic manner.

▶ ▶

5.8 Exercises

1. Find a LU factorization of

 1 2 0

2 1 3

1 2 3

 .

2. Find a LU factorization of

 1 2 3 2

1 3 2 1

5 0 1 3

 .

3. Find a PLU factorization of

 1 2 1

1 2 2

2 1 1

 .

4. Find a PLU factorization of

 1 2 1 2 1

2 4 2 4 1

1 2 1 3 2

 .

5. Find a PLU factorization of

1 2 1

1 2 2

2 4 1

3 2 1

 .

6. Is there only one LU factorization for a given matrix? Hint: Consider the equation(0 1

0 1

)=

(1 0

1 1

)(0 1

0 0

).

7. Here is a matrix and an LU factorization of it.

A =

 1 2 5 0

1 1 4 9

0 1 2 5

 =

 1 0 0

1 1 0

0 −1 1

 1 2 5 0

0 −1 −1 9

0 0 1 14

Use this factorization to solve the system of equations

Ax =

 1

2

3

