Ariā Matrix: Ngā Kaupapa Taketake ki ngā Taupānga
Pendahuluan
He ariā taketake ngā matihiko i roto i te pāngarau, ā, he whānuitia ngā tono i roto i ngā momo mara pērā i te ahupūngao, te ōhanga, te miihini, te pūtaiao rorohiko, me ētahi atu. Ko tēnei hanganga pāngarau he whakaritenga tapawhā rite o ngā tau, ngā huānga rānei i roto i ngā rarangi me ngā pou. I roto i tēnei tuhinga, ka matapakihia e mātou te ariā taketake o ngā matihiko, ō rātou momo, ngā mahi taketake, me ētahi tono nui.
Te Whakamāramatanga o te Matrix
Ko te tikanga, ko te matihiko he huinga tau, he huānga rānei kua whakaritea ki ngā rarangi me ngā pou i roto i te āhua tapawhā rite. Ko te matihiko me ngā rarangi e m me ngā pou e n ka kiia he matihiko m x n. Hei tauira:
\[
A = \begin{pmatrix}
1 me te 2 me te 3
4 me te 5 me te 6
7 & 8 & 9
\end{pmatrix}
\]
He matihiko 3×3 a A nā te mea e 3 ngā rarangi me ngā pou e 3. Ko ngā huānga o te matihiko e tohuhia ana e \( a_{i,j} \), ko i te tohu o te taupū rarangi, ā, ko j te tohu o te taupū pou.
Ngā momo Matrices
Kore Matrix
Ko te matihiko he kore katoa ōna huānga ka kiia he matihiko kore. Ko te tohu e whakamahia whānuitia ana ko O.
\[
O = \begin{pmatrix}
0 me te 0
0 & 0
\end{pmatrix}
\]
Matrix Tuakiri
Ko te matihiko tapawhā kei roto ngā huānga uara kotahi i te whakarara matua (mai i te taha maui o runga ki te taha matau o raro) me ngā kore i ngā wāhi katoa e kiia ana he matihiko tuakiri. Ko te tohu mō te matihiko tuakiri ko I.
\[
I = \begin{pmatrix}
1 me te 0
0 & 1
\end{pmatrix}
\]
Matrix Whakarara
Kāore he huānga o te matihiko whakarara i waho o te whakarara matua. Kāore e taea e ngā huānga o te whakarara matua te kore-kore.
\[
D = \begin{pmatrix}
1 me te 0 me te 0
0 me te 2 me te 0
0 & 0 & 3
\end{pmatrix}
\]
Matrix Whakawhiti
Ko te matihiko whakawhiti he matihiko i whiwhi mā te whakawhiti rarangi mō ngā pou i roto i te matihiko. Hei tauira, mēnā he matihiko A tā tātou:
\[
A = \begin{pmatrix}
1 me te 2
3 & 4
\end{pmatrix}
\]
Kātahi ko te whakawhiti o A (e tohuhia ana e \( A^T \)) ko:
\[
A^T = \begin{pmatrix}
1 me te 3
2 & 4
\end{pmatrix}
\]
Ngā Mahi Matrix
Tāpiritanga Matrix
Ka tāpirihia ngā matihiko e rua mā te tāpiri i ō rāua huānga e rite ana. Hei tauira:
\[
A = \begin{pmatrix}
1 me te 2
3 & 4
\end{pmatrix}, \quad B = \begin{pmatrix}
5 me te 6
7 & 8
\end{pmatrix}
\]
\[
A + B = \begin{pmatrix}
1+5 me te 2+6
3+7 me te 4+8
\end{pmatrix} = \begin{pmatrix}
6 me te 8
10 & 12
\end{pmatrix}
\]
Te Whakareatanga Matrix
Ka taea te whakarea i ngā matihiko e rua A me B mēnā he ōrite te maha o ngā pou i A ki te maha o ngā rarangi i B. Ko te huānga \( c_{i,j} \) o te hua o ngā matihiko C = AB ka tatauhia penei:
\[
c_{i,j} = \sum_{k=1}^{n} a_{i,k} b_{k,j}
\]
Hei tauira:
\[
A = \begin{pmatrix}
1 me te 2
3 & 4
\end{pmatrix}, \quad B = \begin{pmatrix}
5 me te 6
7 & 8
\end{pmatrix}
\]
Ko te hua o \( AB \) ko:
\[
AB = \begin{pmatrix}
1\cdot5 + 2\cdot7 me 1\cdot6 + 2\cdot8
3\cdot5 + 4\cdot7 me 3\cdot6 + 4\cdot8
\end{pmatrix} = \begin{pmatrix}
19 me te 22
43 & 50
\end{pmatrix}
\]
Kaiwhakatau Matrix
Ko te whakatau o tētahi matihiko tapawhā he uara ka taea te whakamahi hei tirotiro i te āheinga hurihuri (te āheinga o te whai whakamuri) o te matihiko. Mō tētahi matihiko 2×2:
\[
A = \begin{pmatrix}
a me b
c me d
\end{pmatrix}
\]
Ko te whakatau ko \( det(A) = ad – bc \).
Matrix Whakamuri
Ko te whakahurihuri o te matihiko A ko te matihiko \( A^{-1} \) kia \( A \cdot A^{-1} = I \), ko I te matihiko tuakiri. He whakahurihuri tō te matihiko A mēnā kāore tōna whakatau i te ōrite ki te kore, ā, mēnā anake kāore.
Tauira o te whakahurihanga o te matihiko 2×2:
\[
A = \begin{pmatrix}
a me b
c me d
\end{pmatrix}
\]
Ko te whakahuri ko:
\[
A^{-1} = \frac{1}{ad – bc} \begin{pmatrix}
d & -b \\
-c me te a
\end{pmatrix}
\]
Taupānga Matrix
Pūnaha o ngā Whārite Raina
E whakamahia whānuitia ana ngā matihiko hei tohu me te whakaoti rapanga i ngā pūnaha whārite rārangi. Hei tauira, ko te pūnaha rārangi:
\[
\begin{ngā take}
2x + 3y = 5
4x + y = 6
\end{ngā take}
\]
ka taea te tuhi ki te puka matihiko:
\[
AX = B
\]
dengan
\[
A = \begin{pmatrix}
2 me te 3
4 & 1
\end{pmatrix}, \quad X = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad B = \begin{pmatrix}
5
6
\end{pmatrix}
\]
Whakairoiro Rorohiko
I roto i ngā whakairoiro rorohiko, ka whakamahia ngā matihiko mō ngā momo panonitanga pēnei i te whakawhiti, te hurihanga, me te tauine o ngā mea i roto i te wāhi toru-ahu. Ka taea te whakaatu i ia panonitanga hei matihiko, ā, mā te whakarea i tēnei matihiko ki ngā taunga o ngā pūwāhi o te mea, ka taea te mahi i te panonitanga mea me te whai hua.
Tātari raraunga
I roto i te tātari raraunga, ka whakamahia ngā matihiko mō ngā kaupapa maha, pērā i te tātari wāhanga matua (PCA) me te wehewehe uara takitahi (SVD). Ka whakamahia te PCA hei whakaiti i te rahi o ngā huinga raraunga nui kia māmā ake ai te tātari, ko te SVD ia hei wehewehe i ngā matihiko ki ngā āhua māmā ake.
Ariā Whatunga
Ka whakamahia hoki ngā matihiko i roto i te ariā whatunga hei tohu i ngā kauwhata. Ko te matihiko pātata tētahi tauira o te matihiko e whakamahia ana hei tohu i ngā whanaungatanga i waenga i ngā pūnga i roto i te kauwhata, hei āwhina i te tātari i ngā whanaungatanga me ngā rerenga i roto i te whatunga.
Whakamutunga
He mea nui te mārama ki ngā ariā taketake o ngā matihiko, ō rātou momo, me ngā mahi e pā ana ki aua mea ki te pāngarau tono. Ko te whānuitanga o ngā whakamahinga o ngā matihiko, mai i ngā pūnaha whārite rārangi ki ngā tikanga tātai i roto i ngā whakairoiro rorohiko, te tātari raraunga, me te ariā whatunga, e whakaatu ana i tō rātou hiranga ki te whakaoti rapanga uaua. Mā te pūtake pakari o ngā ariā matihiko, ka ngāwari ake tā tātou ako i ngā tikanga matatau me ngā tono pāngarau i roto i ngā momo marautanga.
