Ανάστροφος Αντίστροφος

1. EÐdh pinˆkwn
Grammik 'Algebra
Anˆstrofoc pÐnakac
AntÐstrofoc pÐnakac
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
15 OktwbrÐou 2014

2. EÐdh pinˆkwn
Anˆstrofoc - Summetrikìc PÐnakac
Anˆstrofoc tou pÐnaka A eÐnai o pÐnakac pou prokÔptei apo ton
A an kˆnoume tic st lec grammèc kai tic grammèc
st lec kai sumbolÐzetai me AT.

3. EÐdh pinˆkwn
Anˆstrofoc - Summetrikìc PÐnakac
Anˆstrofoc tou pÐnaka A eÐnai o pÐnakac pou prokÔptei apo ton
A an kˆnoume tic st lec grammèc kai tic grammèc
st lec kai sumbolÐzetai me AT.
AT i,j ÆAj,i 8i, j

4. EÐdh pinˆkwn
Anˆstrofoc - Summetrikìc PÐnakac
Anˆstrofoc tou pÐnaka A eÐnai o pÐnakac pou prokÔptei apo ton
A an kˆnoume tic st lec grammèc kai tic grammèc
st lec kai sumbolÐzetai me AT.
AT i,j ÆAj,i 8i, j
Summetrikìc pÐnakac eÐnai Kˆje pÐnakac pou eÐnai Ðsoc me ton
anˆstrofì tou.

5. EÐdh pinˆkwn
Anˆstrofoc - Summetrikìc PÐnakac
Anˆstrofoc tou pÐnaka A eÐnai o pÐnakac pou prokÔptei apo ton
A an kˆnoume tic st lec grammèc kai tic grammèc
st lec kai sumbolÐzetai me AT.
AT i,j ÆAj,i 8i, j
Summetrikìc pÐnakac eÐnai Kˆje pÐnakac pou eÐnai Ðsoc me ton
anˆstrofì tou.
AÆAT ,ai,j Æ aj,i

6. EÐdh pinˆkwn
ParadeÐgmata Summetrik¸n Pinˆkwn
AÆ
2
4
3
5
1 ¡2 3
¡2 6 4
3 4 9

7. EÐdh pinˆkwn
ParadeÐgmata Summetrik¸n Pinˆkwn
AÆ
2
4
3
5 ÆAT Æ
1 ¡2 3
¡2 6 4
3 4 9
2
4
1 ¡2 3
¡2 6 4
3 4 9
3
5

8. EÐdh pinˆkwn
ParadeÐgmata Summetrik¸n Pinˆkwn
AÆ
2
4
3
5 ÆAT Æ
1 ¡2 3
¡2 6 4
3 4 9
2
4
1 ¡2 3
¡2 6 4
3 4 9
3
5
2
1 2 3 4
2 5 6 7
3 6 8 9
4 7 9 10
6664
3
7775
,
2
66664
a1,1 a1,2 ¢ ¢ ¢ a1,n
a1,2 a2,2 ¢ ¢ ¢ a2,n
...
...
. . .
...
a1,n a2,n ¢ ¢ ¢ an,n
3
77775

9. EÐdh pinˆkwn
AntÐstrofoc - Antistrèyimoc PÐnakac
AntÐstrofoc tou pÐnaka A eÐnai o pÐnakac pou sumbolÐzoume me
A¡1, o opoÐoc an pollaplasiajeÐ me ton A mac dÐnei
ton tautotikì pÐnaka.

10. EÐdh pinˆkwn
AntÐstrofoc - Antistrèyimoc PÐnakac
AntÐstrofoc tou pÐnaka A eÐnai o pÐnakac pou sumbolÐzoume me
A¡1, o opoÐoc an pollaplasiajeÐ me ton A mac dÐnei
ton tautotikì pÐnaka.
AA¡1 ÆA¡1AÆ I

11. EÐdh pinˆkwn
AntÐstrofoc - Antistrèyimoc PÐnakac
AntÐstrofoc tou pÐnaka A eÐnai o pÐnakac pou sumbolÐzoume me
A¡1, o opoÐoc an pollaplasiajeÐ me ton A mac dÐnei
ton tautotikì pÐnaka.
AA¡1 ÆA¡1AÆ I
Antistrèyimoc pÐnakac eÐnai kˆje pÐnakac pou èqei antÐstrofo.
AntÐstrofoc tou pÐnaka A eÐnai o pÐnakac pou anaireÐ thn drˆsh
tou en lìgw pÐnaka.

12. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I

13. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D

14. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5
1 0 0
0 6 0
0 0 9

15. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5 D¡1 Æ
1 0 0
0 6 0
0 0 9
2
1 0 0
0 16
64
0
0 0 19
3
75
Ï P pÐnakac antimetˆjeshc )P¡1 Æ

16. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5 D¡1 Æ
1 0 0
0 6 0
0 0 9
2
1 0 0
0 16
64
0
0 0 19
3
75
Ï P pÐnakac antimetˆjeshc )P¡1 Æ P.
Ï Ek,l(p) jemelei¸dhc pÐnakac )
³
Ek,l(p)
´¡1
Æ

17. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5 D¡1 Æ
1 0 0
0 6 0
0 0 9
2
1 0 0
0 16
64
0
0 0 19
3
75
Ï P pÐnakac antimetˆjeshc )P¡1 Æ P.
Ï Ek,l(p) jemelei¸dhc pÐnakac )
³
Ek,l(p)
´¡1
Æ Ek,l(¡p).

18. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5 D¡1 Æ
1 0 0
0 6 0
0 0 9
2
1 0 0
0 16
64
0
0 0 19
3
75
Ï P pÐnakac antimetˆjeshc )P¡1 Æ P.
Ï Ek,l(p) jemelei¸dhc pÐnakac )
³
Ek,l(p)
´¡1
Æ Ek,l(¡p).
E3,1(6) Æ
2
1 0 0 0
0 1 0 0
6 0 1 0
0 0 0 1
6664
3
7775
,

19. EÐdh pinˆkwn
ParadeÐgmata AntÐstrofwn Pinˆkwn
Ï I¡1 Æ I
Ï D diag¸nioc pÐnakac )D¡1 diag¸nioc pÐnakac me diag¸nia
stoiqeÐa ta antÐstrofa antÐstoiqa stoiqeÐa tou D
DÆ
2
4
3
5 D¡1 Æ
1 0 0
0 6 0
0 0 9
2
1 0 0
0 16
64
0
0 0 19
3
75
Ï P pÐnakac antimetˆjeshc )P¡1 Æ P.
Ï Ek,l(p) jemelei¸dhc pÐnakac )
³
Ek,l(p)
´¡1
Æ Ek,l(¡p).
E3,1(6) Æ
2
1 0 0 0
0 1 0 0
6 0 1 0
0 0 0 1
6664
3
7775
,
³
E3,1(6)
´¡1
Æ E3,1(¡6) Æ
2
1 0 0 0
0 1 0 0
¡6 0 1 0
0 0 0 1
6664
3
7775

20. EÐdh pinˆkwn
Upologismìc AntÐstrofou PÐnaka
Na upologisjeÐ o antÐstrofoc enìc dojèntoc pÐnaka A
A
³
A¡1
´
Æ I!Avj Æ ej, j Æ 1,2, . . . ,n
ìpou vj h j-sth st lh tou A¡1 kai ìpou ej h j-sth st lh tou I.
Algìrijmoc
1. LÔnw giˆ j Æ 1, . . . ,n ta grammikˆ sust mata
Avj Æ ej
.
2. Ta vj eÐnai oi antÐstoiqec st lec tou A¡1.