Ngā momo Matrices
Ko te matihiko he whakaritenga o ngā tau, o ngā huānga rānei i roto i ngā rarangi me ngā pou kua whakaritea kia tapawhā rite, kia tapawhā rānei. Ko ngā matihiko he ariā taketake i roto i te pāngarau e whakamahia ana i roto i ngā momo mara pēnei i te ahupūngao, te tatauranga, te pūtaiao rorohiko, me te hangarau. I roto i tēnei tuhinga, ka tūhuratia e mātou ngā momo matihiko e whakamahia whānuitia ana i roto i ngā tono rerekē.
1. Matū Tuakiri
He matihiko tapawhā te matihiko tuakiri me ngā huānga 1 kei te taha matua, me te 0 i ngā wāhi katoa. He maha ngā wā ka tohuhia e te reta "I" me te "E" rānei. Nā ngā āhuatanga o te matihiko tuakiri e rite ai ki te tau 1 i roto i te whakarea noa.
Hei tauira, mō tētahi matihiko tuakiri 3×3, ko te āhua tēnei e whai ake nei:
\[ I = \begin{pmatrix}
1 me te 0 me te 0
0 me te 1 me te 0
0 me te 0 me te 1
\end{pmatrix} \]
He tino whai hua te matihiko tuakiri i roto i ngā mahi taurangi rārangi, inā koa i roto i te tukanga whakaoti rapanga o ngā pūnaha whārite rārangi me te kimi i te whakamuri o tētahi matihiko.
2. Matū Whakarara
He matihiko tapawhā te matihiko whakarara, ko ngā huānga katoa kei waho o te whakarara matua he kore, ā, ko ngā huānga kei runga i te whakarara matua he tau. Ko tōna āhua taketake ko:
\[ D = \begin{pmatrix}
d_1 me te 0 me te 0 \\
0 me te d_2 me te 0 \\
0 me te 0 me te d_3 \\
\end{pmatrix} \]
E whakamahia pinepinetia ana ngā matihiko whakarara i roto i te maha o ngā rauropi pāngarau me ngā tikanga tātai nā te mea he māmā noa iho te tatau, inā koa i roto i te horopaki o te whakarea matihiko.
3. Matrix Kore
Ko te matihiko kore he matihiko e kore katoa ngā huānga. Ka taea e te matihiko kore te tapawhā, te tapawhā hāngai rānei. Ko te tohu noa mō te matihiko kore ko te "0".
Hei tauira, ko tētahi tauira o tētahi matihiko kore 2×3 ko:
\[ 0 = \begin{pmatrix}
0 me te 0 me te 0
0 me te 0 me te 0
\end{pmatrix} \]
He mea nui te matihiko kore i roto i te ariā matihiko hei huānga tuakiri mō te mahi tāpiri matihiko.
4. Matū Āhuarite
He matihiko tapawhā te matihiko taurite, ā, he taurite ngā ihirangi o roto i tōna taha matua. Arā, he ōrite te huānga kei te tūranga (i, j) ki te huānga kei te tūranga (j, i) mō ngā i me ngā j katoa. Nō reira, mēnā he matihiko taurite a \( A \), ko \( A = A^T \), ko \( A^T \) te whakawhiti o \( A \).
Tauira o tētahi matihiko ōrite 3×3:
\[ A = \begin{pmatrix}
2 me te 3 me te 4
3 me te 5 me te 6
4 me te 6 me te 0
\end{pmatrix} \]
He maha ngā wā ka puta mai ngā matihiko ōrite i roto i te maha o ngā raruraru ahupūngao me ngā tatauranga, inā koa i roto i te tātari uara-eigen me te tātaritanga eigenvector.
5. Matrix Anti-Symmetric
Ko te matihiko taurite-kore, ko te matihiko taurite-kore rānei, he matihiko tapawhā kei reira te huānga kei te tūranga (i, j) ko te kino o te huānga kei te tūranga (j, i), ka kiia ko te taurite-kore mēnā ko te A = -A^T.
Tauira o tētahi matihiko taurite-kore 3×3:
\[ A = \begin{pmatrix}
0 me te -2 me te 4
2 me te 0 me te 6
-4 me -6 me te 0
\end{pmatrix} \]
E whakamahia pinepine ana ngā matihiko kore-taurite i roto i te ahupūngao, inā koa i roto i te miihini me te ariā mara.
6. Matariki Ā-Whārite
He matihiko tapawhā te matihiko ā-tikanga (orthogonal matrix) ko te (Q^TQ = I), ko te (Q^T) te whakawhiti o te (Q), ā, ko te (I) te matihiko tuakiri. He āhuatanga tino nui tō ngā matihiko ā-tikanga, arā, ka tiakina te roa o ō rātou whākapū me ngā koki i waenganui i ō rātou whākapū i muri i tēnei panonitanga matihiko.
Tauira o tētahi matihiko orthogonal 2×2:
\[ Q = \begin{pmatrix}
0 me te 1
-1 me te 0
\end{pmatrix} \]
He mea tino nui ngā matriki orthogonal i roto i ngā momo mara o te pāngarau tono, pērā i te tātari raraunga me te āhuahanga rorohiko.
7. Matawhā Tapatoru
Ka wehea ngā matihiko tapatoru ki ngā matihiko tapatoru o runga me ngā matihiko tapatoru o raro. Ko te matihiko tapatoru o runga he matihiko tapawhā kei reira ngā huānga katoa i raro i te hauroki matua he kore. I tetahi atu taha, ko te matihiko tapatoru o raro he kore ngā huānga katoa i runga ake i te hauroki matua.
Te matihiko tapatoru o runga 3×3:
\[ U = \begin{pmatrix}
u_{11} me u_{12} me u_{13} \\
0 me u_{22} me u_{23} \\
0 me te 0 me te u_{33} \\
\end{pmatrix} \]
Te matihiko tapatoru o raro 3×3:
\[ L = \begin{pmatrix}
l_{11} me te 0 me te 0 \\
l_{21} me l_{22} me te 0 \\
l_{31} me l_{32} me l_{33} \\
\end{pmatrix} \]
He mea tino noa ngā matihiko tapatoru i roto i ngā tikanga tau me te arorangi rārangi, inā koa i roto i te wetewete LU me te whakaoti rapanga o ngā pūnaha whārite rārangi.
8. Ngā Matriki Takitahi me ngā Matriki Kore-Takitahi
He matihiko tapawhā te matihiko takitahi, kāore he whakamuri, arā, ko tōna taunga he kore. I tetahi atu taha, he matihiko kore-takitahi he matihiko he whakamuri tōna, arā, kāore tōna taunga i te ōrite ki te kore.
Hei tauira, ko te matihiko 2×2 e whai ake nei he matihiko takitahi nā te mea ko tōna whakatau he kore:
\[ A = \begin{pmatrix}
1 me te 2
2 me te 4
\end{pmatrix} \]
\[ \text{Det}(A) = 1 4 – 2 2 = 0 \]
He mea tino nui te mōhio mēnā he takitahi, he kore-takitahi rānei tētahi matihiko i roto i ngā tono maha, pērā i te whakaoti rapanga o ngā whārite rārangi me ngā tauira ōhanga.
9. Matū Whāiti me te Matū Whāiti
Ko te matihiko tūwhera he matihiko e kore te nuinga o ōna huānga, ko te matihiko kikī ia he iti noa iho, kāore rānei he huānga kore. Ka taea te whakahaere me te rokiroki i ngā matihiko tūwhera he nui ake te whai hua i ērā o ngā matihiko kikī, ā, he tino whai hua ēnei i roto i te rorohiko pūtaiao me te hangarau whatunga.
Tauira o tētahi matihiko 4×4:
\[ S = \begin{pmatrix}
0 me te 0 me te 3 me te 0
0 me te 0 me te 0 me te 4
5 me te 0 me te 0 me te 0
0 me te 6 me te 0 me te 0
\end{pmatrix} \]
He maha ngā wā ka kitea ngā matihiko koretake i roto i ngā momo mara, mai i te ariā kauwhata ki te tātari whatunga rorohiko.
Opaniraa
He mea nui te mārama ki ngā momo matihiko ki te pāngarau me ōna tono. He āhuatanga ahurei ō ngā momo matihiko e whai hua ai i roto i ngā momo mara rerekē. Hei tauira, he māmā ngā matihiko tuakiri me ngā matihiko whakarara engari he mea nui ki ngā tataunga taketake, ko ngā matihiko orthogonal me te whakahaere matihiko taratahi he mea nui ki ngā tataunga uaua ake.
Ehara i te mea he mea whai hua te mōhiotanga ki ēnei momo matihiko i roto i ngā horopaki mātauranga anake, engari he mea nui anō hoki i roto i ngā mahi maha, mai i te pūtaiao raraunga ki te hangarau me te ahupūngao. Hei tāpiri, me mārama ngā ākonga me ngā tohunga ki te whakamahi i ēnei momo matihiko i roto i ā rātou mahi o ia rā.
