Puka Matrix Hauroki
Ko ngā matihiko tētahi o ngā ariā tino nui o te pāngarau, inā koa i roto i te arapūnga rārangi. I roto i ngā momo mara - mai i te ahupūngao me te tatauranga ki te ōhanga ki te pūtaiao rorohiko - ka whakamahia ngā matihiko hei tohu raraunga, ngā pūnaha whārite, ngā panonitanga, me te maha atu. I roto i ngā momo matihiko maha e mōhiotia ana, he tūranga motuhake ngā matihiko whakarara nā te mea he māmā noa iho, engari he kaha ki ngā tatau me ngā tātari. Ka matapakihia e tēnei tuhinga te whakamāramatanga, ngā āhuatanga, te āhua whānui, ngā āhuatanga, me ngā tauira o ngā matihiko whakarara.
Te Mārama ki te Matū Whakarara
He matihiko tapawhā te matihiko whakarara (he rite te maha o ngā rarangi ki te maha o ngā pou) kei reira ngā huānga katoa i waho o te whakarara matua he kore. Ko te whakarara matua he mea hanga mai i te taha maui o runga ki te taha matau o raro, arā, ko ngā huānga kei ngā tūranga \((1,1), (2,2), (3,3)\), me ētahi atu.
Arā, ko ngā huānga anake i te hauroki matua ka taea te kore-kore, ko ngā huānga ia kei waho o te hauroki matua me kore. Ko ngā uara i te hauroki matua ka taea te kore, te kore-kore rānei, i runga i te take.
Hei tauira, ko te matihiko e whai ake nei he matihiko whakarara:
\[
\begin{pmatrix}
4 me te 0 me te 0
0 me te -2 me te 0
0 & 0 & 7
\end{pmatrix}
\]
Kia mōhio ko ngā huānga katoa i tua atu i te 4, -2, me te 7 he kore, nō reira ka tutuki i te matihiko te whakamāramatanga o te matihiko hauroki.
Āhua Whānui o te Matrix Diagonal
I te nuinga o te wā, ka taea te tuhi i tētahi matihiko whakarara o te raupapa \(n \times n\) penei:
\[
D =
\begin{pmatrix}
d_1 me te 0 me te 0 me ngā \cdots me te 0 \\
0 me d_2 me 0 me \cdots me 0 \\
0 me te 0 me te d_3 me ngā \cdots me te 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 me te 0 me te 0 me ngā \cdots me te d_n
\end{pmatrix}
\]
I konei, ko \(d_1, d_2, \ldots, d_n\) ngā huānga o te hauroki matua. Ka taea e ia te noho tūturu, te noho tauoti, te noho matatini rānei, i runga i te horopaki.
He maha hoki ngā wā ka whakamahia ngā tuhi poto:
\[
D = \text{diag}(d_1, d_2, \ldots, d_n)
\]
E kī ana tēnei tuhipoka kei roto i te matihiko \(D\) ngā huānga matua o te hauroki \(d_1\) ki \(d_n\) ā, ko ngā huānga katoa he kore.
Ngā Āhuatanga o te Matrix Diagonal
Ko ētahi āhuatanga e māmā ake ai te mōhio ki ngā matihiko whakarara ko:
1. Te matihiko tapawhā e hiahiatia ana
He rahi tonu te matihiko whakarara \(n \times n\), kāore e taea te tapawhā rite.
2. Me kore ngā huānga kore-whakarara
Me 0 ngā huānga katoa \(a_{ij}\) me te \(i \neq j\).
3. Ngā huānga whakarara kore utu
Ka taea e ngā huānga whakarara \(a_{ii}\) te whai uara (tae atu ki te 0).
4. He take motuhake te matihiko whakarara mō te matihiko tapatoru.
He tapatoru o runga, he tapatoru o raro hoki te matihiko whakarara.
Te Hononga ki te Matrix Tuakiri me te Matrix Tauine
He hononga tata tō ngā matrix hauroki ki ētahi atu momo matrix e rua e puta pinepine ana, arā:
1. Matū Tuakiri
He matihiko whakarara te matihiko tuakiri, ā, ko ngā huānga whakarara katoa he ōrite ki te 1:
\[
Ahau =
\begin{pmatrix}
1 me te 0 me te 0
0 me te 1 me te 0
0 & 0 & 1
\end{pmatrix}
\]
He mea nui tēnei matihiko nā te mea he rite tana mahi ki te tau 1 i roto i te whakarea: mā te whakarea i tētahi atu matihiko ki te matihiko tuakiri kāore e whakarerekē i te matihiko (o te rahi e tika ana).
2. Matū Tauine
He matihiko tauine he matihiko whakarara me ngā huānga whakarara katoa he rite te uara, hei tauira \(k\):
\[
kI =
\begin{pmatrix}
k me te 0 me te 0 \\
0 me te k me te 0 \\
0 me te 0 me te k
\end{pmatrix}
\]
Ara, he āhua motuhake te matihiko tauine o te matihiko whakarara, ā, he āhua motuhake te matihiko tuakiri o te matihiko tauine.
Ngā Āhuatanga Hira o ngā Matrix Diagonal
Mā te māmā o te āhua matihiko whakarara ka hoatu ki a ia ngā āhuatanga e tino māmā ai ngā tataunga.
1. Tāpiri me te Tango
Mena he rite te rahi o ngā matrix hauroki o \(D_1\) me \(D_2\), kāti:
– He matihiko whakarara anō hoki a \(D_1 + D_2\)
– He matihiko whakarara anō hoki a \(D_1 – D_2\)
Nā te mea ko ngā huānga e rite ana anake ka puta te tāpiritanga, ā, ka noho kore ngā huānga katoa kāore i te hauroki.
2. Te Whakareatanga o te Matrix Hauroki
He matrix whakarara anō hoki te hua o ngā matrix whakarara e rua. Mēnā:
\[
D_1 = \text{diag}(a_1, a_2, \ldots, a_n), \quad
D_2 = \text{diag}(b_1, b_2, \ldots, b_n)
\]
Nā reira:
\[
D_1D_2 = \text{diag}(a_1b_1, a_2b_2, \ldots, a_nb_n)
\]
He tino whai hua tēnei nā te mea kāore e hiahiatia kia mahi i te whakareatanga matihiko katoa, he mea uaua tonu.
3. Kaiwhakatau
He tino ngāwari te tatau i te mea whakatau o te matihiko whakarara, arā, ko te hua o ōna huānga whakarara:
\[
\det(D) = d_1 \cdot d_2 \cdot \ldots \cdot d_n
\]
4. Whakamuri
Ka taea te huri ngāwari i tētahi matihiko whakarara, mena he korekore ngā huānga whakarara katoa. Ko te whakahurihanga ko:
\[
D^{-1} = \text{diag}\left(\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_n}\right)
\]
Mena he kore tetahi huānga hauroki, ko te kore te mea whakatau, ā, kāore he whakamuri o te matihiko.
5. Tūnga Matrix
He māmā noa iho hoki ngā taupū o tētahi matihiko whakarara:
\[
D^k = \text{diag}(d_1^k, d_2^k, \ldots, d_n^k)
\]
He tino āwhina tēnei mō te tatau i ngā tauira hihiri me ngā panonitanga auau.
Ngā Tauira o ngā Matrix Hauroki me ngā Matrix Kore-Hauroki
Tauira o tētahi matihiko whakarara:
\[
\begin{pmatrix}
3 me te 0
0 & 5
\end{pmatrix}
\]
Ngā tauira o ngā matihiko kāore i te whakarara (nā te mea he kore ngā huānga kore-whakarara):
\[
\begin{pmatrix}
3 me te 1
0 & 5
\end{pmatrix}
\]
Ahakoa he tapatoru o runga te matihiko, ehara i te matihiko whakarara nā te mea ko te huānga (1,2) he 1, ehara i te 0.
Te Whakahāngaitanga: Te Tahuri i te Matrix ki te Āhua Whakahāngaitanga
Haunga te "matihiko whakarara" hei momo matihiko, kei reira tētahi ariā nui e kiia nei ko te whakarara, arā, ko te tukanga o te huri i tētahi matihiko kua hoatu hei āhua whakarara mā te whakawhiti:
\[
A = PDP^{-1}
\]
ko \(D\) he matihiko whakarara kei roto ngā uara matua, ā, ko \(P\) he matihiko ko ōna pou he eigenvectors. Mēnā ka taea te whakarara i tētahi matihiko, ka māmā ake ngā tātaitanga maha pēnei i te tātai i te tūnga o tētahi matihiko nā te mea he nui te mahi me \(D\).
I roto i te pūtaiao me te hangarau, he maha ngā wā ka whakamahia te whakarara hei whakaoti rapanga i ngā pūnaha rerekētanga, te tātari pumau, te kōpeketanga raraunga, me te tukatuka tohu.
Ngā Whakamahinga o te Matrix Diagonal i te Ao Tūturu
He āhua noa iho te puta mai o ngā matihiko whakarara i roto i ngā momo whakamahinga, hei tauira:
1. Tauine Whakawhiti i roto i ngā Whakairoiro Rorohiko
Hei whakanui, hei whakaiti rānei i tētahi mea motuhake i runga i ngā tuaka \(x\), \(y\), me \(z\), ka whakamahia he matihiko whakarara kei roto i ngā huānga whakarara ngā tauwehe tauine.
2. Te Taurite i roto i ngā Tatauranga
Mena kāore he hononga i waenganui i ngā taurangi matapōkere, ka āhua hauroki te matihiko tauwhitinga nā te mea he kore te tauwhitinga i waenganui i ngā taurangi.
3. Tauira Raina me te Taumaha
I roto i te arotautanga me te ako mīhini, ka whakamahia pinepinetia ngā matihiko whakarara hei matihiko taumaha e tuku ana i ngā whiu rerekē ki ia wāhanga.
Te Katinga
Ko te āhua matihiko whakarara tētahi o ngā hanganga matihiko māmā rawa engari he tino whai hua. Ko te āhua o tēnei matihiko ko te kore o ngā huānga kore-whakarara katoa, engari ka rerekē pea ngā huānga whakarara. Mā tēnei āhua ka māmā ake ngā mahi nui pēnei i ngā whakatau, ngā whakahurihuri, ngā whakarea, me te taupū. Ehara i te mea he mea nui noa iho ngā matihiko whakarara i roto i te arorangi rārangi, engari he whānuitia hoki te whakamahinga i roto i ngā tono o te ao tūturu, mai i ngā tatauranga ki ngā whakairoiro rorohiko.
Ko te mārama ki ngā matihiko whakarara he taahiraa tuatahi nui ki te ako i ngā ariā matatau ake pēnei i ngā uara eigen, ngā eigenvectors, me te whakarara, koinei te ngako o ngā tikanga tātai maha o ēnei rā.
