1. Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141
Math 307
Spring, 2003
Hentzel
Time: 1:10-2:00 MWF
Room: 1324 Howe Hall
Instructor: Irvin Roy Hentzel
Office 432 Carver
Phone
Text: Linear Algebra With Applications,
Second Edition Otto Bretscher

2. Monday, March 24 Chapter 5.3
Page Problems 2,6,8,12,34
Main Idea: Orthonormal vectors make Orthogonal matrices.
Key Words: QR-Factorization, (A B)T = BT AT, Transpose,
VoW = V T W = WT V.
Goal: Learn about orthogonal matrices and the matrix of orthogonal projections.

3. Previous Assignment
Page 199 Problem 14
Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
V1 V2 V3

4. W1 = V1 = 1 7 1
V2oW
W2 = V W1 = = 0
W1oW

5. W3 =
V3oW V3oW
V W W2 = = 1
W1oW W2oW

6. Basis
1/ /Sqrt[2] 0 1/Sqrt[2] 1

7. Page 199 Problem 28
Using paper and pencil, find the QR factorizations of the matrices in Exercises 15 through 28.

8. | 1/10 -1/Sqrt[2] || | | |
| 7/ /Sqrt[2] || 0 Sqrt[2] 0 | = | |
| 1/ /Sqrt[2] || Sqrt[2]| | |
| 7/ /Sqrt[2] || | | 7 7 6|

9. Page 199 Problem 40
Consider an invertible nxn matrix A whose
columns are orthogonal, but not necessarily
orthonormal. What does the QR Factorization
of A look like?

10. Since A is invertible, the columns of A are non
zero.
If A = [ C1 C Cn ] of lengths d1, d2, ... dn
then

11. A is invertible, the columns of A are nonzero.
If A = [ C1 C Cn ] of lengths d1, d2, ... dn
then
| d |
| d |
| |
A = [1/d1 C1 1/d2 C /dn Cn ] | |
| |
| dn|

12. Vectors V1 V2 ... Vn are orthonormal if
Vi o Vj = 0 for i =/= j
and
Vi o Vi = 1 for all i.
Then nxn matrix is called orthogonal if its columns are orthonormal vectors.

13. Properties of orthogonal matrices.
(i) AT A = A AT = I
(ii) A -1 = A T
(iii) | A V | = | V | A preserves lengths.
(iv) If A and B are orthogonal, then AB is orthogonal.
(v) If A is orthogonal, then A -1 is orthogonal.
Proof: Suppose [V1 V2 ... Vn ] is an orthogonal matrix.

14. Then
| V1 T | | | | |
| V2 T | | | | |
| | | V1 V Vn| = | |
| | | | | |
| | | | | |
| Vn T | | | | |

15. We show Part (i).
This is just the statement that
Vi o Vj = 0 if i =/= j and
Vi o Vi = 1 for all i.
This shows that A T A = I.
But then A T is A -1 and so it works on the
other side as well giving A A T = I.

16. Part (ii) is a consequence of Part (i).

17. Part (iii)
| A V | 2 = A V o A V
= (A V) T A V
= V T A T A V
= VT V
= | V | 2

18. Part (iv) If A and B are orthogonal, then
(AB) T (AB) = B T A T A B = I. Thus the
columns of AB are also orthonormal.

19. Part (v) If A is orthogonal, then A A T = I and so
the columns of A T are also orthonormal.

20. The matrix of an orthogonal projection.
Consider a subspace V or R n with orthonormal basis
V1, V2 ... Vm. The matrix of the orthogonal projection
onto V is
| |
A A T where A = | V1 V Vm |

21. Page 209 Example 7.
Find the matrix of the orthogonal projection onto the subspace of R 4 spanned by
| 1 | | 1 |
½ | 1 | ½ | -1 |
| 1 | | -1 |

22. Solution
| ½ ½ | | ½ ½ |
| ½ - ½ || ½ ½ ½ ½ | = | 0 ½ ½ 0 |
| ½ - ½ || ½ - ½ - ½ ½ | | 0 ½ ½ 0 |