Su'aalo Tusaale ah oo Ka Hadlaya Noocyada Matrices-ka
Matrices waa fikrad aasaasi ah oo ku jirta aljabrada toosan waxayna muhiim u yihiin laamaha kala duwan ee sayniska, sida fiisigiska, dhaqaalaha, tirakoobka, iyo injineernimada. Matrices waxay ka kooban yihiin walxo leydi ah oo loo habeeyey saf iyo tiirar. Maqaalkan, waxaan ka hadli doonnaa noocyo kala duwan oo matrices ah, iyo sidoo kale tusaalooyin iyo xalal nooc kasta ah.
1. Matrix-ka Aqoonsiga
Matrix aqoonsi waa matrix laba jibbaaran oo leh 1 curiye oo ku yaal geesoodkiisa ugu weyn (laga bilaabo bidixda sare ilaa midig hoose) iyo 0 curiye oo ka baxsan leexleexadka ugu weyn. Matrix aqoonsi waxaa badanaa lagu tilmaamaa \(I\).
Tusaale:
\[ I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\dhammaad{pmatrix} \]
Su'aal:
Haddii \( A = \begin{pmatrix}
5 & 2 \\
1 & 4
\end{pmatrix} \), hel natiijada ku dhufashada \( A \) iyadoo loo marayo shaxda aqoonsiga \( I \).
Dood:
Matrix-ka \( 2 \times 2 \), matrix-ka aqoonsiga waa:
\[ I = \begin{pmatrix}
1 & 0 \\
0 & 1
\dhammaad{pmatrix} \]
Haddaba, isku dhufashada waa:
\[ AI = \begin{pmatrix}
5 & 2 \\
1 & 4
\end{pmatrix} \begin{pmatrix}
1 & 0 \\
0 & 1
\dhammaad{pmatrix} = \bilaw{pmatrix}
5 & 2 \\
1 & 4
\dhammaad{pmatrix} \]
Natiijadu wali waa shaxda \(A\) lafteeda.
2. Matrix Eber ah
Matrix eber ah waa matrix oo curiyeheedu dhammaantood yihiin 0. Matrix eber ah badanaa waxaa lagu tilmaamaa \(0\).
Tusaale:
\[ 0_2 = \begin{pmatrix}
0 & 0 \\
0 & 0
\dhammaad{pmatrix} \]
Su'aal:
Haddii \(B = \begin{pmatrix}
3 & 7 \\
5 & 9
\end{pmatrix}\), hel natiijada \(B + 0\).
Dood:
Ku dhufashada matrix eber waxay bixisaa natiijo la mid ah matrix-kii asalka ahaa:
\[ B + 0 = \begin{pmatrix}
3 & 7 \\
5 & 9
\end{pmatrix} + \begin{pmatrix}
0 & 0 \\
0 & 0
\dhammaad{pmatrix} = \bilaw{pmatrix}
3 & 7 \\
5 & 9
\dhammaad{pmatrix} \]
3. Matrix-ka Shaashadda
Matrix-ka toosan waa matrix laba jibbaaran oo dhammaan walxaha ka baxsan leexleexadka ugu weyn ay yihiin 0. Walxaha ku yaal leexleexadka ugu weyn way kala duwanaan karaan, laakiin walxaha ka baxsan leexleexadka ugu weyn waa inay dhammaantood ahaadaan 0.
Tusaale:
\[ D = \begin{pmatrix}
6 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 8
\dhammaad{pmatrix} \]
Su'aal:
Matrix-kan soo socdaa ma yahay matrix toosan?
\[ C = \begin{pmatrix}
5 & 0 \\
0 & 6
\dhammaad{pmatrix} \]
Dood:
C waa matrix laba jibbaaran oo leh curiyeyaal ka baxsan geesoodka ugu weyn dhammaantood waa 0. Sidaa darteed, \( C \) runtii waa matrix toosan.
4. Matrix-ka Iskeelka
Matrix-ka scalar waa qaab gaar ah oo ka mid ah matrix-ka leexleexadka ah kaas oo dhammaan walxaha leexleexadka ugu muhiimsan ay siman yihiin. Matrix-ka scalar waxaa loo qaadan karaa inuu yahay isku-dhufashada scalar ee shaxda aqoonsiga.
Tusaale:
\[ S = \begin{pmatrix}
4 & 0 \\
0 & 4
\dhammaad{pmatrix} \]
Su'aal:
Caddee in matrix \(T\) ee hoose uu yahay matrix scalar ah:
\[ T = \begin{pmatrix}
7 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & 7
\dhammaad{pmatrix} \]
Dood:
Matrix-ku waa matrix toosan oo dhammaan walxaha geesoodka ugu muhiimsan ay yihiin 7. Sidaa darteed, \(T\) waa matrix scalar ah.
5. Matrix-ka Isku-dhafan
Matrix siman waa matrix laba jibbaaran oo la mid ah transpose-kiisa. Taas macnaheedu waa in walxaha isku dheelitiran ee ku saabsan leexleexadka ugu muhiimsan ay siman yihiin, taas oo ah, \(A_{ij} = A_{ji}\) mid kasta oo \(i\) iyo \(j\).
Tusaale:
\[ A = \begin{pmatrix}
2 & 1 & 3 \\
1 & 4 & 5 \\
3 & 5 & 6
\dhammaad{pmatrix} \]
Su'aal:
Hubi in shaxda soo socota ay tahay jaantus siman:
\[ B = \begin{pmatrix}
1 & 2 \\
2 & 3
\dhammaad{pmatrix} \]
Dood:
Beddelka \(B\) waa:
\[ B^T = \bilaw{pmatrix}
1 & 2 \\
2 & 3
\dhammaad{pmatrix} \]
Maadaama \( B = B^T \), markaas \( B \) waa matrix siman.
6. Matrix Saddex-xagal ah
Matrices-ka saddex-xagalku waxay ku yimaadaan laba nooc: saddex-xagal sare iyo saddex-xagal hoose. Matrix-ka saddex-xagal sare wuxuu leeyahay dhammaan walxaha ka hooseeya xariiqda ugu weyn oo la mid ah 0, halka matrix-ka saddex-xagal hoose uu leeyahay dhammaan walxaha ka sarreeya xariiqda ugu weyn oo la mid ah 0.
Tusaale: Saddexagal Sare:
\[ U = \begin{pmatrix}
2 & 3 & 4 \\
0 & 5 & 6 \\
0 & 0 & 7
\dhammaad{pmatrix} \]
Tusaale ahaan: Saddexagal Hoose:
\[ L = \begin{pmatrix}
8 & 0 & 0 \\
5 & 6 & 0 \\
3 & 4 & 2
\dhammaad{pmatrix} \]
Su'aal:
Go'aami noocyada shaxda soo socota:
\[ C = \begin{pmatrix}
1 & 2 \\
0 & 3
\dhammaad{pmatrix} \]
Dood:
Maadaama dhammaan walxaha ka hooseeya leexleexadka ugu weyn ay yihiin 0, markaas \(C \) waa matrix saddexagal sare ah.
7. Matrix-ka Orthogonal
Matrix-ka toosan waa matrix laba jibbaaran \(A\) oo buuxiya isla'egta \(A^TA = AA^T = I \), halkaas oo \(A^T \) uu yahay wareejinta \(A\) iyo \(I\) uu yahay matrix-ka aqoonsiga.
Tusaale:
\[ Q = \begin{pmatrix}
1/2 & \sqrt{3}/2 \\
\sqrt{3}/2 & -1/2
\dhammaad{pmatrix} \]
Su'aal:
Hubi in matrices-ka hoose ay yihiin kuwo toosan:
\[ P = \begin{pmatrix}
0 & 1 \\
1 & 0
\dhammaad{pmatrix} \]
Dood:
Marka hore waxaan xisaabinaynaa wareejinta \(P\):
\[ P^T = \bilaw{pmatrix}
0 & 1 \\
1 & 0
\dhammaad{pmatrix} \]
Kadib waxaan xisaabineynaa \( P^TP \):
\[ P^TP = \bilaw{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \begin{pmatrix}
0 & 1 \\
1 & 0
\dhammaad{pmatrix} = \bilaw{pmatrix}
1 & 0 \\
0 & 1
\dhammaad{pmatrix} = I \]
Maadaama \( P^TP = I \), markaas \(P\) waa matrix orthogonal ah.
Markaan fahanno noocyada kala duwan ee matrices-ka iyo astaamahooda, waxaan si fudud u go'aamin karnaa xalalka dhibaatooyinka xisaabeed ee kala duwan ee ku lug leh matrices-ka. Nooc kasta oo matrix ah wuxuu leeyahay sifooyin gaar ah oo loo isticmaali karo codsiyada sayniska iyo farsamada ee kala duwan.
