4-Mavzu: Matritsalar va ularning ayrim хоssalari

Izoh: Dioganal element, uchburchak matritsa, dioganal matritsa, birlik



Ta’rif. Ikkita o’zaro vertikal chiziq bilan ajratilgan, satrlari soni bir xil bo’lgan ixtiyoriy ikkita matritsadan iborat C=(A|B) ko’rinishidagi matritsa kengaytirilgan matritsa deyiladi.

misol
matritsa, skalyar matritsa, simmetrik matritsa va kososimmetrik matritsa tushunchalari faqatgina kvadrat matritsalar uchungina o’rinli.

15. 
    

    𝜆𝑎11			𝜆𝑎12	⋯	𝜆𝑎1𝑛
    C=𝜆𝐴=[^𝜆𝑎^21 ⋯			𝜆𝑎22 ⋯	⋯ ⋯	𝜆𝑎2𝑛] ⋯
    𝜆𝑎𝑚1			𝜆𝑎𝑚2	⋯	𝜆𝑎𝑚𝑛
    	3	1	2
    Misol.	^A=^[1	5	6^]	va		𝜆 = 4	𝜆𝐴 ni	aniqlang.
    	1	6	4

    
    
    
    Ta’rif. A=(𝑎_𝑖j)mxn matritsaning haqiqiy songa ko’paytymasi deb elementlari: 𝑐_𝑖j = 𝜆𝑎_𝑖j (i=1,2,…,n.) kabi aniqlangan C=(𝑐_𝑖j )mxn matritsaga aytiladi.
    

4	8
20	24^]
24	16

3 1 2 4 · 3 4 · 1 · 4 · 2 12
^{𝜆𝐴=4· [}1 5 6^{]= [} 4 · 1 4 · 5 4 · 6 ^{] = [} 4
1 6 4 4 · 1 4 · 6 4 · 4 4
Teorema: Itiyoriy o’lchamli A matritsa, 𝜆 𝑣𝑎 𝜇 –haqiqiy sonlar uchun quyidagi munosabatlar o’rinli:

1. 
   assotsiativlik: 𝜆 ⁽𝜇𝐴⁾ = (𝜆 𝜇)𝐴
   
2. 
   sonlarni qo’shish (ayirish)ga nisbatan distributivlik : (𝜆 ± 𝜇)A= 𝜆𝐴 ± 𝜇𝐴
   

Matritsalarni o’zaro qo’shish (ayirish).

16. 
    Ta’rif. Ikkita A va B matritsaning yig’indi (ayirma)sining natijasi C matritsa bo’lib, uning elementlari 𝑐_𝑖j =𝑎_𝑖j + (−)𝑏_𝑖jkabi aniqlanadi.
    

Matritsani qo’shish (ayirish) amali quydagi xossalarga ega:

• 
  kommutativlik: A±B=B±A;
  
• 
  assotsiativlik: (A±B)±C=A±(B±C)
  
• 
  qo’shish (ayirish)ga nisbatan distributivlik: λ(A±B)=λA±λB Bu yerda A, B,va C bir xil o’lchamli matritsalar, λ o’zgarmas son
  

1 2 3 1 3 4
^{Misol. A=[}2 1 4^{] ; B=[}5 7 8^{] matritsalar uchun 2A+B ni hisoblang.}

3 2 3
1 2 3 2
1 2 4

4	6
2	8^]
4	6
	2 + 1	4 + 3	6 + 4	3	7	10

^{2A=2 · [}2 1 4^{] = [}4
3 2 3 6
2 4 6 1 3 4
^2A+B=[4 2 8^{] +[}5 7 8^]=[4 + 5 2 + 7 8 + 8^]=[9 9 16^]

6 4 6
1 2 4
6 + 1 4 + 2 6 + 4
7 6 10

1	2	3	1	3	4
^Misol.^A=^[2	1	4^]^;	^B=^[5	7	8^]	matritsalar uchun 2A-B ni hisoblang.
3	2	3	1	2	4

1 2 3 2 4 6
^{2A=2 · [}2 1 4^{] = [}4 2 8^{] 2A-}

3 2 3

2	4
^B=^[4	2
6	4

6 1 3 4
6 4 6
2 − 1 4 − 3 6 − 4
1 1 2

8^{] -[}5 7 8^]=[4 − 5 2 − 7 8 − 8^]=[−1 −5 0^]

6 1 2 4
6 − 1 4 − 2 6 − 4
5 2 2

Matritsalar uchun ko’rsatilgan chiziqli amallarni bajaring.
1) A=‖^{1 −1 −3}‖ , B=‖ ^{0 3 2}‖ 3A-2B=?
2 1 5 −1 4 1

3	5	7		1	2	4
^2)^A=^[2	−1	0	] , B=[	2	3	−2^]	2A+4B=?

4 3 −6 −1 0 1

Izoh: Turli o’lchamli matritsalarni qo’shib (ayirib) bo’lmaydi. Matritsani matritsaga ko’paytirish.

17. 
    Ta’rif A=(𝑎_𝑖j) va B=(𝑏_𝑖j) matritsalarning ko’paytmasidan iborat bo’lgan C=A∙B=(𝑐_𝑖j ) matritsaning elementlari quyidagi formula yordamida
    

aniqlanadi:

𝑘−1
𝑐_𝑖j =^∑𝑛 𝑎_𝑖𝑘 ∙𝑏_𝑘j=𝑎_𝑖1𝑏_1j+𝑎_𝑖2𝑏_2j+…+𝑎_𝑖𝑛𝑏_𝑛j (2)
(2) formuladan ko’rinib turibdiki, A∙B ko’paytirish amali faqatgina A matritsaning ustunlari soni va B matritsaning satirlari soni o’zaro teng bo’lgandagina amalga oshiriladi.
Izoh:Ko’paytirish amalida ko’paytmadagi matritsalarning joylashgan o’rni ahamiyatli.Shu sababli matritsalar uchun o’ngdan va chapdan ko’paytirish qoidalari mavjud.

^𝑎11 ^𝑎12 ^𝑎13
^𝑏11 ^𝑏12 ^𝑏13

Misol. A=[_{𝑎21 𝑎22 𝑎23}] , B=[^{𝑏21 𝑏22 𝑏23}] bo’lsa A·B ni toping.
^𝑏31 ^𝑏32 ^𝑏33

^𝑎11 ^𝑎12 ^𝑎13
^𝑏11 ^𝑏12 ^𝑏13

C=A·B=[_{𝑎21 𝑎22 𝑎23}] [^{𝑏21 𝑏22 𝑏23}]=
^𝑏31 ^𝑏32 ^𝑏33
_[ ^𝑎11^𝑏11 ^{+ 𝑎}12^𝑏21 ^{+ 𝑎}13^𝑏31 ^𝑎11^𝑏12 ^{+ 𝑎}12^𝑏22 ^{+ 𝑎}13^𝑏32 ^𝑎11^𝑏13 ^{+ 𝑎}12^𝑏23 ^{+ 𝑎}13^𝑏33 _]
^𝑎21^𝑏11 ^{+ 𝑎}22^𝑏21 ^{+ 𝑎}23^𝑏31 ^𝑎21^𝑏12 ^{+ 𝑎}22^𝑏22 ^{+ 𝑎}23^𝑏32 ^𝑎21^𝑏13 ^{+ 𝑎}22^𝑏23 ^{+ 𝑎}23^𝑏33
Demak, (2×3) o’lchovli A matritsani (3×3) o’lchovli B matritsaga ko’paytirilganda (2×3) o’lchovli C matritsa hosil bo’ladi.

A matritsaning o’lchamlari (m×n), B matritsaning o’lchamlari (n×q) bo’lsa, C=A·B matritsaning o’lchami (m×q) bo’ladi.

1. 
   Misol: Berilgan A va B matritsalar uchun C=AB ni aniqlang.
   

1
^A=[4^{] va B=[}2 4 1^{] uchun AB va BA ko’paytmani aniqlang.}
3

	1		1 · 2	1 · 4	1 · 1	2	4	1
Yechish:	^AB=^[4^]^·^[2	4	1^]^=^[4 · 2	4 · 4	4 · 1^]^=	^[8	16	4^]
	3		3 · 2	3 · 4	3 · 1	6	12	3

1
^BA=[2 4 1^{] · [}4^{] = 2 · 1 + 4 · 4 + 1 · 3 = 2 + 16 + 3 = 21}
3

2. 
   

   −2 va	B=	2 ^[−3	3 6	−4 0	5 1 ^]	bo’lsa, C=AB ni
   5		3	−5	6	7

   
   
   
   Misol: A= ^{1 3}
   

aniqlang.


C=AB=
[
4 −1


1 3 −2
]
2 3 −4 5

^[_{4 −1 5} ^{] · [}−3 6 0 1 ^{] =}
3 −5 6 7
_[1 · 2 − 3 · 3 − 2 · 3 1 · 3 + 3 · 6 + 2 · 5 −1 · 4 + 3 · 0 − 2 · 6 1 · 5 + 3 · 1 − 2 · 7_]
4 · 2 + 1 · 3 + 5 · 3 4 · 3 − 1 · 6 − 5 · 5 −4 · 4 − 1 · 0 + 5 · 6 4 · 5 − 1 · 1 + 5 · 7

]=
_=[ 2 − 9 − 6 3 + 18 + 10 −4 + 0 − 12 5 + 3 − 14
8 + 3 + 15 12 − 6 − 25 −16 − 0 + 30 20 − 1 + 35
_[−13 31 −16 − 6_]
26 −19 14 54
Berilgan A, B matritsalar uchun A· 𝐵 amallarni bajaring.

1 −3 0
0 −1 3

^{1) A=[}_{2 5 1}^{] , 𝐵 = [}3 5 2^]
4 −2 1

	1	3	1	2	1	0
2)	^A=^[2	0	4^]^,	^B=^[1	−1	2^]
	1	2	3	3	2	1

Teorema: Matritsalarni ko’paytirish amali quyidagi xossalarga ega:

1. 
   Matritsalarni ko’paytirish amali matritsalarni qo’shish amaliga nisbatan distrubutiv,ya’ni agar A(B+C) va (A+B)C mavjud bo'lsa u holda:A(B+C)=AB+AC va (A+B)C=AC+BC munosabatlar o’rinli.
   
2. 
   Matritsalarni ko’paytirish amali assotsiativ,ya’ni agar AB va(AB)C ko’paytmalar mavjud bo’lsa,u holda ABvaA(BC) ko’paytmalar ham mavjud bo’ladi va quyidagi munosabat o’rinli:
   

(AB)C=A(BC)

3. 
   Agar AB ko’paytma mavjud bo’lsa,ixtiyoriy α o’zgarmas son uchun quyidagi tenglik o’rinli:
   

α(AB)=(αA)B=A(αB)