Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα
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Transcript of Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα
![Page 1: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/1.jpg)
GRAMMIK'H 'ALGEBRA27h di�lexh:Probolèc
23 NoembrÐou 2012
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Probol se eujeÐa tou Rn
Na brejeÐ h probol p tou b ep�nw sthn eujeÐa
pou orÐzei to a
mNa brejeÐ to plhsièstero sto b shmeÐo p thc
eujeÐac pou orÐzei to a
To a dièrqetai apo thn arq twn axìnwn.
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Probol se eujeÐa tou Rn
![Page 4: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/4.jpg)
Probol se eujeÐa tou Rn
![Page 5: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/5.jpg)
H probol p enìc dianÔmatocb ∈ Rn se mia eujeÐa a ∈ Rn poupern�ei apo to 0
- eÐnai h p = aTb
aT aa
- me antÐstoiqo pÐnaka probol c
P = aaT
aT aI που είναι συμμετρικός και τάξης 1
![Page 6: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/6.jpg)
H probol p enìc dianÔmatocb ∈ Rn se mia eujeÐa a ∈ Rn poupern�ei apo to 0
- eÐnai h p = aTb
aT aa
- me antÐstoiqo pÐnaka probol c
P = aaT
aT aI που είναι συμμετρικός και τάξης 1
![Page 7: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/7.jpg)
H probol p enìc dianÔmatocb ∈ Rn se mia eujeÐa a ∈ Rn poupern�ei apo to 0
- eÐnai h p = aTb
aT aa
- me antÐstoiqo pÐnaka probol c
P = aaT
aT a
I που είναι συμμετρικός και τάξης 1
![Page 8: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/8.jpg)
H probol p enìc dianÔmatocb ∈ Rn se mia eujeÐa a ∈ Rn poupern�ei apo to 0
- eÐnai h p = aTb
aT aa
- me antÐstoiqo pÐnaka probol c
P = aaT
aT aI που είναι συμμετρικός και τάξης 1
![Page 9: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/9.jpg)
Par�deigma
- Probol tou
123
sto
111
- PÐnakac probol c sto
111
kai
sto
[cos θsin θ
]
![Page 10: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/10.jpg)
Par�deigma
- Probol tou
123
sto
111
- PÐnakac probol c sto
111
kai
sto
[cos θsin θ
]
![Page 11: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/11.jpg)
Par�deigma
- Probol tou
123
sto
111
- PÐnakac probol c sto
111
kai
sto
[cos θsin θ
]
![Page 12: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/12.jpg)
Efarmog
UpologÐste x tètoio ¸ste Ax = b kai x /∈ R(A).
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Efarmog
UpologÐste x tètoio ¸ste Ax = b
kai x /∈ R(A).
![Page 14: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/14.jpg)
Efarmog
UpologÐste x tètoio ¸ste Ax = b kai x /∈ R(A).
![Page 15: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/15.jpg)
Efarmog
UpologÐste x tètoio ¸ste Ax = b kai x /∈ R(A).
![Page 16: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/16.jpg)
El�qista Tetr�gwna
An Ax = b kai x /∈ R(A) tìte
miaprosèggish thc lÔshc x eÐnai hlÔsh y tou sust matoc Ay = pìpou p h probol tou b ston R(A).
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El�qista Tetr�gwna
An Ax = b kai x /∈ R(A) tìte miaprosèggish thc lÔshc x eÐnai hlÔsh y tou sust matoc Ay = pìpou p h probol tou b ston R(A).
![Page 18: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/18.jpg)
El�qista Tetr�gwna
Ax = b
⇒ ATAx = ATb⇒
x =(ATA
)−1ATb
Pr�gmati AT (Ax − b) = 0
Upìjesh: oi st lec tou A eÐnaigrammik� anex�rthtec.
![Page 19: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/19.jpg)
El�qista Tetr�gwna
Ax = b⇒ ATAx = ATb
⇒
x =(ATA
)−1ATb
Pr�gmati AT (Ax − b) = 0
Upìjesh: oi st lec tou A eÐnaigrammik� anex�rthtec.
![Page 20: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/20.jpg)
El�qista Tetr�gwna
Ax = b⇒ ATAx = ATb⇒
x =(ATA
)−1ATb
Pr�gmati AT (Ax − b) = 0
Upìjesh: oi st lec tou A eÐnaigrammik� anex�rthtec.
![Page 21: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/21.jpg)
El�qista Tetr�gwna
Ax = b⇒ ATAx = ATb⇒
x =(ATA
)−1ATb
Pr�gmati AT (Ax − b) = 0
Upìjesh: oi st lec tou A eÐnaigrammik� anex�rthtec.
![Page 22: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/22.jpg)
El�qista Tetr�gwna
Ax = b⇒ ATAx = ATb⇒
x =(ATA
)−1ATb
Pr�gmati AT (Ax − b) = 0
Upìjesh: oi st lec tou A eÐnaigrammik� anex�rthtec.
![Page 23: Διάλεξη 27η - Προβολές και ελάχιστα τετράγωνα](https://reader031.fdocuments.net/reader031/viewer/2022020101/559a96f21a28ab046f8b477b/html5/thumbnails/23.jpg)
Par�deigma
1 4
1 5
0 6
x =
456