Mehlala ea Lipotso tse Buisanang ka Mefuta ea Matrice
Li-matrice ke mohopolo oa motheo ho algebra e otlolohileng 'me li bohlokoa makaleng a fapaneng a saense, joalo ka fisiks, moruo, lipalo-palo le boenjiniere. Li-matrice li na le likarolo tse khutlonnetsepa tse hlophisitsoeng ka mela le likholomo. Sehloohong sena, re tla buisana ka mefuta e fapaneng ea li-matrice, hammoho le mehlala le litharollo bakeng sa mofuta o mong le o mong.
1. Setšoantšo sa Boitsebiso
Matrix ea boitsebiso ke matrix e sekwere e nang le likarolo tse 1 ho daegonale ea eona e kholo (ho tloha holimo ka letsohong le letšehali ho ea tlase ka ho le letona) le likarolo tse 0 ho tloha ho daegonale e kholo. Matrix ea boitsebiso hangata e bontšoa ke \(I\).
Contoh:
\[ I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 le 1
\end{pmatrix} \]
Potso:
Haeba \( A = \begin{pmatrix}
5 le 2 \\
1 & 4
\end{pmatrix} \), fumana sephetho sa ho atisa \( A \) ka matrix ya boitsebiso \( I \).
Puisano:
Bakeng sa matrix \( 2 \makgetlo a 2 \), matrix ya boitsebiso ke:
\[ I = \begin{pmatrix}
1 le 0 \\
0 & 1
\end{pmatrix} \]
Kahoo, katiso ke:
\[ AI = \begin{pmatrix}
5 le 2 \\
1 & 4
\end{pmatrix} \begin{pmatrix}
1 le 0 \\
0 & 1
\end{pmatrix} = \begin{pmatrix}
5 le 2 \\
1 & 4
\end{pmatrix} \]
Sephetho e ntse e le matrix \(A\) ka boyona.
2. Matrix ea Lefela
Matrix ea lefela ke matrix eo likarolo tsa eona kaofela e leng 0. Matrix ea lefela hangata e bontšoa ke \(0\).
Contoh:
\[ 0_2 = \begin{pmatrix}
0 le 0 \\
0 & 0
\end{pmatrix} \]
Potso:
Haeba \(B = \begin{pmatrix}
3 le 7 \\
5 & 9
\end{pmatrix}\), fumana sephetho \(B + 0\).
Puisano:
Ho atisa ka matrix ea lefela ho fana ka sephetho se tšoanang le matrix ea pele:
\[ B + 0 = \begin{pmatrix}
3 le 7 \\
5 & 9
\end{pmatrix} + \begin{pmatrix}
0 le 0 \\
0 & 0
\end{pmatrix} = \begin{pmatrix}
3 le 7 \\
5 & 9
\end{pmatrix} \]
3. Matrix e otlolohileng
Matrix e otlolohileng ke matrix e sekwere eo ho yona dielemente tsohle tse ka ntle ho daegonale e kgolo di leng 0. Dielemente tse ho daegonale e kgolo di ka fapana, empa dielemente tse ka ntle ho daegonale e kgolo kaofela di lokela ho ba 0.
Contoh:
\[ D = \begin{pmatrix}
6 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 le 8
\end{pmatrix} \]
Potso:
Na matrix e latelang ke matrix e otlolohileng?
\[ C = \begin{pmatrix}
5 le 0 \\
0 & 6
\end{pmatrix} \]
Puisano:
C ke matrix e sekwere e nang le dielemente tse ka ntle ho daegonale e kgolo kaofela e le 0. Ka hona, \( C \) ehlile ke matrix e taegonale.
4. Scalar Matrix
Matrix ea scalar ke mofuta o ikhethang oa matrix e otlolohileng moo likarolo tsohle tsa dayagonale e ka sehloohong li lekanang. Matrix ea scalar e ka nkoa e le multiplier ea scalar holim'a matrix ea boitsebiso.
Contoh:
\[ S = \begin{pmatrix}
4 le 0 \\
0 & 4
\end{pmatrix} \]
Potso:
Paka hore matrix \(T\) e ka tlase ke matrix ea scalar:
\[ T = \begin{pmatrix}
7 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 le 7
\end{pmatrix} \]
Puisano:
Matrix \(T\) ke matrix e otlolohileng moo likarolo tsohle tsa daegonale e ka sehloohong e leng 7. Ka hona, \(T\) ke matrix e otlolohileng.
5. Matrix e Tšoanang
Matrix e lekanang ke matrix e sekwere e lekanang le transpose ea eona. Sena se bolela hore likarolo tse lekanang tse mabapi le daegonale e kholo lia lekana, ke hore, \(A_{ij} = A_{ji}\) bakeng sa \(i\) le \(j\) e 'ngoe le e 'ngoe.
Contoh:
\[ A = \begin{pmatrix}
2 & 1 & 3 \\
1 & 4 & 5 \\
3 & 5 le 6
\end{pmatrix} \]
Potso:
Hlahloba hore na matrix e latelang ke matrix e lekanang:
\[ B = \begin{pmatrix}
1 le 2 \\
2 & 3
\end{pmatrix} \]
Puisano:
Phetoho ea \(B\) ke:
\[ B^T = \begin{pmatrix}
1 le 2 \\
2 & 3
\end{pmatrix} \]
Kaha \( B = B^T \), joale \( B \) ke matrix e lekanang.
6. Matrix e khutlotharo
Matrices e khutlotharo e tla ka mefuta e 'meli: e khutlotharo e ka holimo le e khutlotharo e ka tlase. Matrix e khutlotharo e ka holimo e na le likarolo tsohle tse ka tlase ho daegonale e kholo e lekanang le 0, ha matrix e khutlotharo e ka tlase e na le likarolo tsohle tse ka holimo ho daegonale e kholo e lekanang le 0.
Mohlala oa Khutlotharo e ka Holimo:
\[ U = \begin{pmatrix}
2 & 3 & 4 \\
0 & 5 & 6 \\
0 & 0 le 7
\end{pmatrix} \]
Mohlala oa Khutlotharo e ka Tlase:
\[ L = \begin{pmatrix}
8 & 0 & 0 \\
5 & 6 & 0 \\
3 & 4 le 2
\end{pmatrix} \]
Potso:
Khetha mefuta e latelang ea matrix:
\[ C = \begin{pmatrix}
1 le 2 \\
0 & 3
\end{pmatrix} \]
Puisano:
Kaha dielemente tsohle tse ka tlase ho daegonale e kgolo ke 0, jwale \( C \) ke matrix e ka hodimo ya kgutlotharo.
7. Matrix ea Orthogonal
Matrix e otlolohileng ke matrix e sekwere \(A\) e kgotsofatsang equation \( A^TA = AA^T = I \), moo \( A^T \) e leng transpose ya \(A\) mme \(I\) e le matrix ya boitsebiso.
Contoh:
\[ Q = \begin{pmatrix}
1/2 le \sqrt{3}/2 \\
\sqrt{3}/2 le -1/2
\end{pmatrix} \]
Potso:
Netefatsa hore na matrices e ka tlase e na le orthogonal:
\[ P = \begin{pmatrix}
0 le 1 \\
1 & 0
\end{pmatrix} \]
Puisano:
Taba ea pele re bala phetoho ea \(P\):
\[ P^T = \begin{pmatrix}
0 le 1 \\
1 & 0
\end{pmatrix} \]
Ebe re bala \( P^TP \):
\[ P^TP = \begin{pmatrix}
0 le 1 \\
1 & 0
\end{pmatrix} \begin{pmatrix}
0 le 1 \\
1 & 0
\end{pmatrix} = \begin{pmatrix}
1 le 0 \\
0 & 1
\end{pmatrix} = I \]
Kaha \( P^TP = I \), joale \(P\) ke matrix e otlolohileng.
Ka ho utloisisa mefuta e fapaneng ea matrices le litšobotsi tsa eona, re ka fumana litharollo habonolo mathateng a fapaneng a lipalo a amanang le matrices. Mofuta o mong le o mong oa matrices o na le thepa e ikhethang e ka sebelisoang lits'ebetsong tse fapaneng tsa saense le tsa tekheniki.
