Orthogonality and orthonormality Inner product 1 Definition - - PowerPoint PPT Presentation

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Orthogonality and orthonormality Inner product 1 Definition - - PowerPoint PPT Presentation

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Orthogonality and orthonormality Inner product 1 Definition (inner product) Let V be a vector space a function , : V V C is said to be an inner (dot) product if 1. u , w = w , u 2. u

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SLIDE 1

Orthogonality and

  • rthonormality
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SLIDE 2

Inner product

1

Definition (inner product)

Let V be a vector space a function · , · : V × V → C is said to be an inner (dot) product if 1. u , w = w , u

  • 2. α

u , w = α u , w 3. v + u , w = v , w + u , w 4. u , u ≥ 0 where u , u = 0 if and only if u =

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SLIDE 3

Example

2

· , · : C3 × C3 → C

  • u ,

w =   u1 u2 u3   ,   w1 w2 wd  

  • =
  • u1

u2 u3

 w1 w2 w3   = uT w = u1 w1 + u2 w2 + u3 w3

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SLIDE 4

Example

3

  −2i + 3 i + 1     1 i + 4 −4i + 3     −3i + 2 2i + 3 i − 1  

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SLIDE 5

Example

4

· , · : C3 × C3 → C

  • u ,

w =   u1 u2 u3   ,   w1 w2 wd  

  • =
  • u1

u2 u3

 2 1 5     w1 w2 w3   = uT M w = 2 u1 w1 + 1 u2 w2 + 5 u3 w3

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SLIDE 6

Example

5

· , · : Cd × Cd → C

  • u ,

w =

    u1 u2 . . . ud      ,      w1 w2 . . . wd     

  • =
  • u1

u2 . . . ud

    w1 w2 . . . wd      = uT w =

d

  • i=1

ui wi

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SLIDE 7

Example

6

· , · : Cd × Cd → C

  • u ,

w =

    u1 u2 . . . ud      ,      w1 w2 . . . wd     

  • =
  • u1

u2 . . . ud

  • M

     w1 w2 . . . wd      = uT M w M = M∗ and positive definite

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SLIDE 8

Norm

7

Definition (Norm)

Let V be a vector space a function · : V → [0, ∞) is said to be a norm if 1. u = 0 if and only if u =

  • 2. α

u = |α| u 3. u + w ≤ u + w

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SLIDE 9

Example

8

· : Cd → R

  • u

=

  u1 . . . ud   

  • =
  • u1 u1 + · · · + ud ud

=

  • u1

. . . ud

  u1 . . . ud    =

  • uT u
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SLIDE 10

Example

9

  −2i + 3 i + 1     1 i + 4 −4i + 3     −3i + 2 2i + 3 i − 1  

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SLIDE 11

Example

10

Let · , · : V × V → C define · : V → R

  • u =
  • u ,

u Recall

  • u ,

w =

  • u1

u2 u3

 2 1 5     w1 w2 w3   = 2 u1 w1 + 1 u2 w2 + 5 u3 w3

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SLIDE 12

Orthogonal vectors

11

Definition

Let V be a vector space and · , · be an inner product on

  • V. We say

u, w ∈ V are orthogonal if

  • u ,

w = 0

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SLIDE 13

Example

12

    i + 1 1 −i + 1 i         5i + 1 5i + 6 −i − 7 −6i + 1         −i + 3 i + 1 i         −4i − 2 i + 6 3i + 4 −i + 6    

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SLIDE 14

Example

13

  • u ,

w = uT   2 1 5   w   −2i + 3 i + 1     1 i + 4 −4i + 3     −3i + 2 2i + 3 i − 1     25i − 5 60i + 60 26  

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SLIDE 15

Orthogonality and independence

14

Definition

A set of vectors U = { u1, u2, . . . , uk} is called orthogonal if i = j ⇒

  • ui ,

uj

  • = 0

Theorem

Let U = { u1, u2, . . . , uk} be an orthogonal set of vectors. Then U is linearly independent.

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SLIDE 16

Gram-Schmidt procedure

15

Theorem

Let W = { w1, w2, . . . , wk} be a set of linearly independent

  • vectors. Define U = {

u1, u2, . . . , uk} by

  • ui =

wi − u1 , wi

  • u1 ,

u1 u1 − u2 , wi

  • u2 ,

u2 u2 − · · · − ui−1 , wi

  • ui−1 ,

ui−1 ui−1 Then the vectors U are an orthogonal set of vectors and W = U

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SLIDE 17

Example

16

  1 1 1     2     1 3     1 i + 1 1     −i 1 i + 1     i i  

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SLIDE 18

Orthonormal set

17

Definition

An orthogonal set of vectors U = { u1, u2, . . . , uk} is

  • rthonormal if for all i
  • ui = 1