Te Whakamahinga o ngā Whakatau i roto i te Āraipa
He ariā nui te whakatau i roto i te arorangi rārangi e puta pinepine ana i roto i ngā kōrero mō ngā matihiko. Ahakoa te āhua nei he mahi tātai noa iho i te tīmatanga, he hohonu ake te tikanga o te whakatau: ka āwhina i a tātou ki te mārama ki ngā āhuatanga o tētahi matihiko, ki te whakatau mēnā he otinga ahurei tō tētahi pūnaha whārite, ki te tatau i te whakahuri o tētahi matihiko, tae atu ki te whakamārama i ngā panonitanga rārangi mā te āhuahanga. Ka matapakihia whānuitia e tēnei tuhinga te whakamahinga o ngā whakatau i roto i te arorangi, mai i tō rātou whakamāramatanga ki ō rātou tono matua.
Te Mārama ki ngā Mea Whakatau
I ngā kupu māmā noa iho, ko te whakatau he tau tauine e hono ana ki tētahi matihiko tapawhā (he matihiko he rite te maha o ngā rarangi me ngā pou). Ko ngā whakatau ka tautuhia mō ngā matihiko tapawhā anake, hei tauira, 2×2, 3×3, me ētahi atu. Ko te tuhi whakatau ka tuhia hei det(A) mā te whakamahi rānei i tētahi pae poutū, hei tauira, |A|.
Mō tētahi matihiko 2×2:
\[
A = \begin{pmatrix} a me b \\ c me d \end{pmatrix}
\]
kātahi ko te mea whakatau ko:
\[
\det(A) = ad – bc
\]
He tohu nui te uara o te whakatau: ki te kore te whakatau, ka "takitahi" te matihiko (kāore he whakamuri); ki te kore i te kore, ka "kore-takitahi" te matihiko (he whakamuri).
Ngā Kaiwhakatau me ngā Pūnaha o ngā Whārite Raina
Ko tētahi o ngā whakamahinga e akohia pinepinetia ana mō ngā whakatau i roto i te pāngarau ko te whakaoti rapanga i ngā pūnaha whārite rārangi. Whakaarohia te pūnaha whārite rārangi e whai ake nei i roto i ngā taurangi e rua:
\[
\begin{ngā take}
toki + mā = e \\
cx + dy = f
\end{ngā take}
\]
Ka taea te tuhi i tēnei pūnaha ki te puka matihiko:
\[
\begin{pmatrix} a me b \\ c me d \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}
=
\begin{pmatrix} e \\ f \end{pmatrix}
\]
Mena ko te whakatau o te matihiko tauwehenga \(\det(A) = ad – bc \neq 0\), kāti he otinga ahurei tā te pūnaha. I tetahi atu taha, ki te ōrite te whakatau ki te kore, kāti ka taea e te pūnaha te whai otinga mutunga kore, ka kore rānei he otinga, i runga i te ōritetanga o ngā whārite.
I tēnei horopaki, ka mahi te mea whakatau hei "mea whakatau" mēnā ka taea te whakaoti motuhake i te pūnaha, kāore rānei.
Te Ture a Cramer
Ko te whakamahinga o ngā whakatau whakatau hei whakaoti i tētahi pūnaha whārite e kiia ana ko te Ture a Cramer. E kī ana tēnei ture mō tētahi pūnaha whārite rārangi me te maha o ngā taurangi rite ki ngā whārite, ka taea te whiwhi i te otinga mā te whakataurite i ētahi whakatau whakatau.
Mō te pūnaha 2×2 i runga ake nei, ko te otinga ko:
\[
x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}
\]
ko \(A_x\) he matihiko kua whakakapia tōna pou tuatahi ki te pūmau (e, f), ā, ko \(A_y\) he matihiko kua whakakapia tōna pou tuarua ki te pūmau.
He pai te tikanga a Cramer mō te mārama ki ngā ariā, ahakoa i roto i ngā tātaitanga tau nui ka whakamahia pinepinetia te tikanga whakakore Gaussian nā te mea he pai ake.
Ngā Whakatau mō te Tatau i ngā Whakamuri o te Matrix
He mea nui anō hoki ngā whakatau i te kimi i te whakahuri o tētahi matihiko. Ko te whakahuri o tētahi matihiko A, e tohuhia ana ko \(A^{-1}\), ka noho noa mēnā ko \(\det(A) \neq 0\).
Mō tētahi matihiko 2×2:
\[
A^{-1} = \frac{1}{ad – bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]
E whakaatu mārama ana tēnei tātai kei roto te taupū i te mea whakatau. Mēnā he kore te mea whakatau, kāore e taea te wehewehe, ā, kāore hoki te whakamuri e puta.
Mō ngā matihiko nunui ake, pērā i te 3×3, ka taea te kimi i te whakahuri mā te whakamahi i te tikanga tāpiri, e uru ana hoki te kaiwhakatau o te matihiko iti (iti) me te kaiwhakatau tahi. Mā tēnei ka whakanuia ko te kaiwhakatau kei te ngako o te ariā whakahuri.
Ngā Kaiwhakatau me ngā Huringa Raina
I roto i te arorangi rārangi, ka tirohia ngā matihiko hei whakaaturanga o ngā panonitanga rārangi, pērā i ngā panonitanga i roto i te papa (2D) i te wāhi rānei (3D). Ka taea te whakamārama i te whakatau hei tauine mō te panonitanga o te horahanga, te rōrahi rānei e puta mai ana i te panonitanga.
– Mō tētahi matihiko 2×2, ko te uara o |det(A)| e tohu ana i te whakarea horahanga.
– Mō tētahi matihiko 3×3, ko te uara o |det(A)| e tohu ana i te whakarea rōrahi.
Hei tauira, mēnā ko te \(\det(A) = 3\), ka toru ngā wā ka nui ake te horahanga o tētahi āhua papatahi kua hurihia e A. Mēnā ko te \(\det(A) = -2\), ka rua ngā wā ka nui ake te horahanga, engari ko te tohu kino e tohu ana i te huringa o te aronga (hei tauira, te whakaata).
Nā tēnei tikanga āhuahanga, he nui ake te whakatau i te mea he taputapu tatau noa iho—e whakamārama ana i te āhua o ngā panonitanga mā te hinengaro noa.
Ngā Whakatau i te Tirotiro i te Whakawhirinakitanga Raina
I roto i te pāngarau, he mea nui te ariā o te whakawhirinakitanga rārangi, inā koa ka kōrerohia ngā whārite me ngā pūtake. Ka taea te whakamahi i ngā whakatau hei whakatau mēnā he motuhake rārangi te huinga whārite.
Hei tauira, e toru ngā whārite i roto i te wāhi 3D ka taea te whakaaro he pou o tētahi matihiko 3×3. Mena ehara i te kore te whakatau o te matihiko, ka tū motuhake ngā whārite e toru, ā, ka waiho hei pūtake mō te wāhi 3D. Mena he kore te whakatau, ka whakawhirinaki hāngai ngā whārite, arā, ka taea te whakaatu i tētahi o ngā whārite hei huinga rārangi o ētahi atu.
He mea whai hua i roto i ngā mara maha, pērā i te tātari mokowā whārite, te arotautanga, te ahupūngao, me te rorohiko.
Ngā Kaiwhakatau me te Horahanga/Rōrahi me ngā Wetere
Haunga te whakamārama i ngā panonitanga, ka taea hoki te whakamahi i ngā whakatau whakatau hei tatau tika i ngā horahanga me ngā rōrahi mā te whakamahi i ngā whārite.
– Ka taea te tatau i te horahanga o tētahi whakarara i hangaia e ngā whārite e rua \(u\) me \(v\) i roto i te papa mā te uara tino o te kaiwhakatau o te matihiko ko ōna pou ko u me v.
– Ka taea te tatau i te rōrahi o tētahi porowhita whakarara i hangaia e ngā whārite e toru i roto i te wāhi 3D mā te uara tino o te kaiwhakatau o tētahi matihiko 3×3, ko ōna pou ko ngā whārite e toru.
Ara, he taputapu ine āhuahanga whai hua, huatau hoki te mea whakatau.
Ngā Āhuatanga Hira o ngā Kaiwhakatau i roto i te Āraipa
I roto i te mahi taurangi, ka whakamahia ngā whakatau whakatau me ngā āhuatanga e whai ake nei:
1. det(AB) = det(A)det(B)
He mea nui tēnei i roto i te maha o ngā taunakitanga me ngā tātaitanga.
2. det(A^T) = det(A)
Kāore te mea whakatau e rerekē mēnā ka whakawhitia te matihiko.
3. Ki te whakareatia tētahi rarangi (pou rānei) ki te k, ka whakareatia hoki te taunga whakatau ki te k.
4. Ki te whakawhitihia ngā rarangi e rua, ka huri te tohu o te kaiwhakatau.
5. Mena he ōrite ngā rarangi e rua, he taurea rānei ngā rarangi tetahi ki tetahi, ko te whakatau = 0.
Mā ēnei āhuatanga ka māmā ake te whakamāmā i ngā whakatau whakatau me te kore e tatau i te tīmatanga.
Whakamutunga
Ko te whakamahinga o ngā whakatau i roto i te taurangi e kapi ana i te whānuitanga o ngā wāhanga nui: te whakatau i te noho o ngā otinga ahurei ki ngā pūnaha whārite rārangi, te whakamahi i te Ture a Cramer, te tatau i ngā whakahurihanga matihiko, te tātari i ngā panonitanga rārangi, te whakamātautau i te motuhaketanga rārangi, me te tatau i ngā horahanga me ngā rōrahi mā te āhuahanga. Ko ngā whakatau he ariā e hono ana i te taurangi ki te āhuahanga, ā, e whakarato ana i tētahi huarahi tere ki te aromatawai i te hanganga me ngā āhuatanga o tētahi matihiko.
Mā te mārama pai ki ngā whakatau taurite ka āwhina i a koe ki te whakapakari i tō māramatanga ki te taurangi rārangi whānui, nā te mea he maha ngā kaupapa matatau—pērā i ngā uara taurite, te whakahāngaitanga, me te whakarerekētanga o te pūtake—e ahu mai ana hoki i tēnei ariā. Mā te mōhio ki ngā whakatau taurite ka hoatu he māramatanga matua ki a koe mō te taurangi me te pāngarau o ēnei rā.
