Te Hanganga Whakawhitiwhiti mā te Whakamahi i ngā Matrices
Pendahuluan
He ariā matua te tito whakawhiti i roto i te arorangi rārangi me te āhuahanga, e whakamahia whānuitia ana i roto i ngā momo mara pūtaiao me te hangarau, pērā i te whakairoiro rorohiko, te ahupūngao, me te miihini. I roto i tēnei tuhinga, ka ruku hohonu atu tātou ki te tito whakawhiti mā te whakamahi i ngā matihiko. He taputapu kaha, he ngāwari hoki ngā matihiko hei whakahaere i ngā mahi whakawhiti, ā, mā te mārama ki tēnei ariā ka taea e tātou te whakamahi i aua mea i roto i ngā horopaki uaua.
Matrix i roto i te Hurihanga
Whakamāramatanga me te Whakaaturanga
He whakaritenga tapawhā rite te matihiko o ngā huānga kei roto ko ngā rarangi me ngā pou. Mā te pāngarau, ka tohuhia te matihiko hei A me ngā huānga aᵢⱼ, ko te i hei rarangi, ā, ko te j hei pou. Hei tauira, ka taea te whakaatu i te matihiko 2×2 penei:
\[
\mathbf{A} = \begin{pmatrix}
he_{11} me he_{12} \\
he_{21} me he_{22}
\end{pmatrix}
\]
I roto i te horopaki o ngā panonitanga rārangi, ka whakamahia ngā matihiko hei whakarerekē i ngā taunga o ngā pūwāhi i te wāhi. Hei tauira, ka taea te whakaatu i te panonitanga o te pūwāhi (x, y) mā te matihiko rārangi penei:
\[
\begin{pmatrix}
x' \\
y '
\end{pmatrix}
=
\mathbf{A}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
Ngā Momo Whakawhitinga Matrix
He maha ngā momo panonitanga taketake ka taea te whakamahi mā te whakamahi i ngā matihiko, tae atu ki:
1. Whakamāoritanga: Ahakoa kāore e taea te whakaatu i te whakamāoritanga hei matihiko rārangi, ka taea te whakahaere i te whakamāoritanga mā te whakamahi i ngā matihiko ōrite.
2. Hurihanga: Ka taea te whakaatu i te hurihanga o tētahi pūwāhi i te papa xy mā te koki \(\theta\) ki te taha matau mā te matihiko hurihanga penei:
\[
\mathbf{R}(\theta) =
\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta me \cos\theta
\end{pmatrix}
\]
3. Tauine: Ka whakanui, ka whakaiti rānei te matihiko tauine i tētahi pūwāhi. Ko te matihiko tauine i ngā taha e rua ko:
\[
\mathbf{S}(s_x, s_y) =
\begin{pmatrix}
s_x me te 0 \\
0 me te s_y
\end{pmatrix}
\]
4. Kutikuti: Ka nekehia e tēnei panonitanga tētahi pūwāhi ki te taha kotahi. Ka taea te whakaatu i te matihiko kutikuti i ngā taha e rua penei:
\[
\mathbf{H}(k_x, k_y) =
\begin{pmatrix}
1 me te k_x \\
k_y me te 1
\end{pmatrix}
\]
Te Hanganga o te Hurihanga
Ko te tito whakawhiti ko te whakamahinga ā-muri o ngā whakawhiti e rua, neke atu rānei, ki tētahi pūwāhi, ki tētahi mea rānei. I roto i te āhua matihiko, ka whakaatuhia te tito whakawhiti hei whakarea matihiko.
Ariā Taketake
Mena kei a tātou ngā panonitanga rārangi e rua e tohuhia ana e ngā matihiko \(\mathbf{A}\) me \(\mathbf{B}\), ko te hanganga o ngā panonitanga e rua \(\mathbf{C}\) ko te hua o ngā matihiko e rua:
\[
\mathbf{C} = \mathbf{A} \times \mathbf{B}
\]
Kātahi ka taea te whakamahi i te panonitanga \(\mathbf{C}\) hei huri i ngā pūwāhi, i ngā mea rānei.
Hei tauira, me kī ka hurihia e tātou mā te \(\theta_1\) ā, ka hurihia e tātou mā te \(\theta_2\). Ko te matihiko whakawhiti katoa ko:
\[
\mathbf{C} = \mathbf{R}(\theta_2) \times \mathbf{R}(\theta_1)
\]
I tēnei wā, ka taea te whakangawari i te hua o te whakareatanga o te matihiko hurihuri mā te whakamahi i ngā āhuatanga pākoki.
Te Whakatinanatanga i roto i ngā Whakairoiro Rorohiko
I roto i ngā whakairoiro rorohiko, he maha ngā wā ka whakamahia ngā panonitanga tito hei whakarerekē i te āhua o ngā mea i roto i te ao whakairoiro. Me kī tātou e hiahia ana ki te whakarerekē i te rahi o tētahi mea, kātahi ka huri. Ko te panonitanga tuatahi ko te matihiko tauine \(\mathbf{S}\) ā, ko te tuarua ko te matihiko hurihanga \(\mathbf{R}\):
\[
\mathbf{C} = \mathbf{R}(\theta) \times \mathbf{S}(s_x, s_y)
\]
Kātahi ka whakareatia ia pūwāhi o te mea ki te matihiko \(\mathbf{C}\) hei whiwhi i ngā taunga hou, kua whakanuia, kua hurihia hoki.
Tauira Hanga
Hei mārama ake i tēnei tukanga, me titiro tātou ki tētahi tauira taipitopito o te hanganga o ngā panonitanga i roto i ngā taahiraa e rua:
1. Whakamahia he tauine rua (s_x = 2, s_y=2) i te pūwāhi (1, 1)
2. Hurihia te pūwāhi tauine ka puta mā te 90 nekehanga ki te taha maui.
Ko te whakaaturanga pāngarau ko:
1. Matū tauine \(\mathbf{S}\):
\[
\mathbf{S} =
\begin{pmatrix}
2 me te 0
0 & 2
\end{pmatrix}
\]
Ko te pūwāhi (1, 1) i muri i te whakapūtanga ka noho hei:
\[
\begin{pmatrix}
2 me te 0
0 & 2
\end{pmatrix}
\begin{pmatrix}
1
1
\end{pmatrix}
=
\begin{pmatrix}
2
2
\end{pmatrix}
\]
2. Te whārite hurihuri \(\mathbf{R}\) mā te 90 nekehanga:
\[
\mathbf{R}(90^\circ) =
\begin{pmatrix}
0 me te -1
1 & 0
\end{pmatrix}
\]
Kātahi ka hurihia te pūwāhi hua o te tauine ki:
\[
\begin{pmatrix}
0 me te -1
1 & 0
\end{pmatrix}
\begin{pmatrix}
2
2
\end{pmatrix}
=
\begin{pmatrix}
-2
2
\end{pmatrix}
\]
Nō reira, ko te hua whakamutunga o te hanganga panoni ko te pūwāhi (-2, 2).
Whakamutunga
He ariā nui te tito o ngā panonitanga mā te whakamahi i ngā matihiko i roto i te pāngarau tono me ngā tono mahi maha. Mā te mārama ki te mahi a te whakarea matihiko me te tito, ka taea e tātou te mahi ngāwari i ngā panonitanga uaua ki ngā mea āhuahanga. He mea nui tēnei ariā i roto i ngā mara pēnei i te whakairoiro rorohiko, te ahupūngao, me te hangarau, e whakarato ana i te turanga pakari mō te mahi me ngā panonitanga rārangi i roto i ngā wāhi maha-ahu.
Kua kapi i tēnei tuhinga ētahi ariā taketake mō ngā matihiko me ngā panonitanga, me te whakamahinga o aua hanganga. Mā te māramatanga hōhonu ki te hanganga panonitanga matihiko, ka taea e tātou te whakatau i ngā raruraru panonitanga maha e pā ana ki a tātou i roto i te pūtaiao me te hangarau.
