Lesson 14 The Gram-Schmidt Process and QR Factorizations

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1 Lesson 14 The Gram-Schmidt Process and QR Factorizations Math 21b March 12, 2007 Announcements Get your midterm from me if you haven t yet. Homework for March 14: 5.2: 6,14,22,34,40,44* Problem Session Wednesdays, 7-8 PM in SC 101b Office hours: Monday 2-4, Tuesday 3-5 in SC 323 (resuming today)

2 The problem Given a basis B for a subspace V of R n, replace it with an orthonormal basis A having the same span.

3 The problem Given a basis B for a subspace V of R n, replace it with an orthonormal basis A having the same span. Why? Because orthonormal bases are just plain better.

4 Why orthonormal bases are just plain better No need to check for linear independence (automatic from orthogonality) Coordinates are easy just dot: v u 1 v u 2 [ v] A =. v u m Orthogonal projections are easy just dot: proj V ( x) = x = ( x u 1 ) u 1 + ( x u 2 ) u ( x u m ) u m

5 The geometric idea In fact, we will do a little better: Given a basis v 1, v 2,..., v m for a subspace V of R n, replace it with an orthonormal basis u 1, u 2,..., u m such that span( v 1, v 2,..., v k ) = span( u 1, u 2,..., u k ) for each k between 1 and m.

6 The geometric idea In fact, we will do a little better: Given a basis v 1, v 2,..., v m for a subspace V of R n, replace it with an orthonormal basis u 1, u 2,..., u m such that span( v 1, v 2,..., v k ) = span( u 1, u 2,..., u k ) for each k between 1 and m. The process works like this: Scale v 1 to make a unit vector u 1. Choose the rest of the u k to be the (normalized) perpendicular part of the orthogonal decomposition of v k relative to span( v 1, v 2,..., v k 1 ).

7 The geometric idea, in pictures v 2 v 1

8 The geometric idea, in pictures v 2 u 1 v 1

9 The geometric idea, in pictures v 2 u 1 v 1

10 The geometric idea, in pictures v 2 v 2 v 1 v 2 u 1

11 The geometric idea, in pictures v 2 v 2 u 2 v 1 v 2 u 1

12 The geometric idea, in pictures v 2 u 2 v 1 u 1

13 The geometric idea, in 3D v 3 v 2 v 1

14 The geometric idea, in 3D v 3 v 1 u 1 v 2

15 The geometric idea, in 3D v 3 v 1 u 1 v 2

16 The geometric idea, in 3D v 2 v 3 v 1 u 1 v 2 v 2

17 The geometric idea, in 3D v 2 v 3 u 2 v 1 u 1 v 2 v 2

18 The geometric idea, in 3D v 3 u 2 v 2 v 1 u 1

19 The geometric idea, in 3D v 3 v 3 u 2 v 2 v 1 u 1 v 3

20 The geometric idea, in 3D v 3 v 3 u 3 u 2 v 2 v 1 u 1 v 3

21 The geometric idea, in 3D v 3 u3 u 2 v 2 v 1 u 1

22 The algebraic idea Scale v 1 to make a unit vector u 1.

23 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1

24 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2.

25 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2

26 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u

27 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u

28 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u This is known as the Gram-Schmidt process for orthonormalization.

29 Worksheet Do Problems 1 3

30 QR Factorizations

31 When are the spans of two sets the same? When is span( v 1, v 2,..., v m ) = span( u 1, u 2,..., u m )?

32 When are the spans of two sets the same? When is span( v 1, v 2,..., v m ) = span( u 1, u 2,..., u m )? If each of the v i s can be written as a linear combination of the u i s and vice versa. v 1 = c 11 u 1 + c 21 u c m1 u m v 2 = c 12 u 1 + c 22 u c m2 u m. v m = c 1m u 1 + c 2m u c mm u m

33 Writing it all in terms of matrices, we have [ v1 c 11 c 12 c 1m ] [ ] c 21 c 22 c 2m v 2... v m = u1 u 2... u m c m1 } c m2 {{ c mm } S

34 Writing it all in terms of matrices, we have [ v1 c 11 c 12 c 1m ] [ ] c 21 c 22 c 2m v 2... v m = u1 u 2... u m c m1 } c m2 {{ c mm } S The matrix S is the change of basis matrix from ( v 1,..., v m ) to ( u 1,..., u m )

35 From Gram-Schmidt to QR Suppose an n m matrix M has rank m, so its columns v 1,..., v m are linearly independent. Let u 1,..., u m be the orthonormal basis of span( v 1,..., v m ) = image M gotten by the Gram-Schmidt process, and let Q = [ u 1... u m ]. Then M = QR for some invertible matrix R.

36 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1

37 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1 v 2 = v 2 ( v 2 u 1 ) u 1 and u 2 = v 2 v 2, so v 2 = ( u 1 v 2 ) u 1 + v 2 u 2

38 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1 v 2 = v 2 ( v 2 u 1 ) u 1 and u 2 = v 2 v 2, so v 2 = ( u 1 v 2 ) u 1 + v 2 u 2 And so on: v k = ( u 1 v k ) u 1 + ( u 2 v k ) u ( u k 1 v k ) u k 1 + v k u k

39 So v 1 u 1 v 2 u 1 v 3 u 1 v m 0 v 2 u 2 v 3 u 2 v m R = 0 0 v 3... u 3 v m v m That is, R is upper triangular. u i v j if i < j r ij = v i = ui v i if i = j 0 if i > j

40 Example Find the QR factorization of M = [ ]

41 Example Find the QR factorization of M = Solution We have v 1 = [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = [ 1/ ] 2. 1/ 2

42 Example Find the QR factorization of M = Solution We have v 1 = [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 ]

43 Example Find the QR factorization of M = Solution We have v 1 = Therefore [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = r 22 = v 2 [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 = 1 u 2 = 1 [ 1/ ] v 2 2 = 2 r 22 1 /. 2 ]

44 Example Find the QR factorization of M = Solution We have v 1 = Therefore [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = r 22 = v 2 [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 = 1 u 2 = 1 [ 1/ ] v 2 2 = 2 r 22 1 /. 2 Putting this all together, [ ] [ 1 1 1/ 2 1/ ] [ 2 2 1/ ] M = = 1 0 1/ 2 1 / / = QR 2 ]

45 Example (Worksheet, #4) Find the QR Factorization of

46 Example (Worksheet, #4) Find the QR Factorization of Solution 1/2 1/2 1 /2 1 / /2 1/2 Q = 1 /2 1 /2 1/2 1/2 1/2 1/2 1/2 R = / /2 1/2 1 /2 1/2 1 / /2

47 Who cares? A lot of important things are easier to calculate once you have the QR factorization, especially if M is square. It turns out Q 1 = Q T, so M 1 = R 1 Q T and R 1 is easier to calculate than M 1. It turns out det Q = ±1, so det M = ± det R, and det R is very easy to calculate (more later) The eigenvalues of M are easier to calculate when M is factored as QR.

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