Orthogonality and orthonormality Inner product 1 Definition - PowerPoint PPT Presentation

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

Example 2 � · , · � : C 3 × C 3 → C �     u 1 w 1 �  , � � u , � w � = u 2 w 2    u 3 w d   w 1 � � = u 1 u 2 u 3 w 2   w 3 u T w = u 1 w 1 + u 2 w 2 + u 3 w 3 =

Example 3       − 2 i + 3 1 − 3 i + 2 i + 4 2 i + 3 0       i + 1 − 4 i + 3 i − 1

Example 4 � · , · � : C 3 × C 3 → C �     u 1 w 1 �  , � � u , � w � = u 2 w 2    u 3 w d     w 1 2 0 0 � � = u 1 u 2 u 3 0 1 0 w 2     0 0 5 w 3 u T M w = 2 u 1 w 1 + 1 u 2 w 2 + 5 u 3 w 3 =

Example 5 � · , · � : C d × C d → C     u 1 w 1 � � u 2 w 2     � � u , � w � =  ,  .   .  . .     . .    u d w d   w 1 w 2   � � = u 1 u 2 . . . u d  .  .   .   w d d u T w = � = u i w i i = 1

Example 6 � · , · � : C d × C d → C     u 1 w 1 � � u 2 w 2     � � u , � w � =  ,  .   .  . .     . .    u d w d   w 1 w 2   � � = u 1 u 2 . . . u d M  .  .   .   w d u T M w M = M ∗ and positive definite =

Norm 7 Definition (Norm) Let V be a vector space a function � · � : V → [ 0 , ∞ ) is said to be a norm if u = � 1. � � u � = 0 if and only if � 0 2. � α� u � = | α |� � u � 3. � � u + � w � ≤ � � u � + � � w �

Example 8 � · � : C d → R � �   u 1 � � � . � . � � u � = �   � . � �   � � u d � � � = u 1 u 1 + · · · + u d u d �   � u 1 � . � . � � = u 1 . . . u d  .  �   � u d � u T u =

Example 9       − 2 i + 3 1 − 3 i + 2 i + 4 2 i + 3 0       i + 1 − 4 i + 3 i − 1

Example 10 Let � · , · � : V × V → C define � � � � � u ,� � · � : V → R u � = u � Recall     2 0 0 w 1 � � � � u , � w � = u 1 u 2 u 3 0 1 0 w 2     0 0 5 w 3 = 2 u 1 w 1 + 1 u 2 w 2 + 5 u 3 w 3

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

Example 12         i + 1 5 i + 1 − i + 3 − 4 i − 2 1 5 i + 6 i + 1 i + 6                 − i + 1 − i − 7 i 3 i + 4         − 6 i + 1 − i + 6 i 0

Example 13   2 0 0 w � = u T  w � � u , � 0 1 0  0 0 5       − 2 i + 3 1 − 3 i + 2 0 i + 4 2 i + 3       i + 1 − 4 i + 3 i − 1   25 i − 5 60 i + 60   26

Orthogonality and independence 14 Definition A set of vectors U = { � u 1 ,� u 2 , . . . ,� u k } is called orthogonal if � � � i � = j ⇒ u i ,� = 0 u j Theorem Let U = { � u 1 ,� u 2 , . . . ,� u k } be an orthogonal set of vectors. Then U is linearly independent.

Gram-Schmidt procedure 15 Theorem Let W = { � w 1 , � w 2 , . . . , � w k } be a set of linearly independent vectors. Define U = { � u 1 ,� u 2 , . . . ,� u k } by w i − � � u 1 , � w i � u 1 − � � u 2 , � w i � u 2 − · · · − � � u i − 1 , � w i � � u i = � u 1 � � u 2 � � u i − 1 � � u i − 1 � � u 1 ,� � � u 2 ,� � � u i − 1 ,� Then the vectors U are an orthogonal set of vectors and � W � = � U �

Example 16       1 0 1 1 2 0       1 0 3       − i 1 0 i + 1 1 i       i + 1 1 i

Orthonormal set 17 Definition An orthogonal set of vectors U = { � u 1 ,� u 2 , . . . ,� u k } is orthonormal if for all i � � u i � = 1