Ngā Tauira Pātai e Matapaki ana i ngā Momo Matrices
He ariā taketake ngā matihiko i roto i te arorangi rārangi, ā, he mea nui hoki i roto i ngā momo peka pūtaiao, pērā i te ahupūngao, te ōhanga, te tatauranga, me te hangarau. Ko ngā matihiko he mea hanga mai i ngā huānga tapawhā rite kua whakaritea ki ngā rarangi me ngā pou. I roto i tēnei tuhinga, ka matapakihia e mātou ngā momo matihiko, me ngā tauira me ngā otinga mō ia momo.
1. Matū Tuakiri
He matihiko tuakiri he matihiko tapawhā kei reira he 1 ngā huānga i tōna hauroki matua (mai i te taha maui o runga ki te taha matau o raro) me te 0 ngā huānga i waho o te hauroki matua. Ko te matihiko tuakiri e tohuhia ana e \(I\).
Tauira:
\[ I_3 = \begin{pmatrix}
1 me te 0 me te 0
0 me te 1 me te 0
0 & 0 & 1
\end{pmatrix} \]
Pātai:
Mena \( A = \begin{pmatrix}
5 me te 2
1 & 4
\end{pmatrix} \), kimihia te hua o te whakareatanga \( A \) ki te matihiko tuakiri \( I \).
Kōrero:
Mō te matihiko \( 2 \times 2 \), ko te matihiko tuakiri ko:
\[ I = \begin{pmatrix}
1 me te 0
0 & 1
\end{pmatrix} \]
Nō reira, ko te whakareatanga ko:
\[ AI = \begin{pmatrix}
5 me te 2
1 & 4
\end{pmatrix} \begin{pmatrix}
1 me te 0
0 & 1
\end{pmatrix} = \begin{pmatrix}
5 me te 2
1 & 4
\end{pmatrix} \]
Ko te hua tonu ko te matihiko \(A\) tonu.
2. Matrix Kore
Ko te matihiko kore he matihiko e 0 katoa ana ngā huānga. Ko te matihiko kore te tohu matua o te \(0\).
Tauira:
\[ 0_2 = \begin{pmatrix}
0 me te 0
0 & 0
\end{pmatrix} \]
Pātai:
Mena \(B = \begin{pmatrix}
3 me te 7
5 & 9
\end{pmatrix}\), kimihia te hua \(B + 0\).
Kōrero:
Ko te whakareatanga ki te matihiko kore ka puta he hua ōrite ki te matihiko taketake:
\[ B + 0 = \begin{pmatrix}
3 me te 7
5 & 9
\end{pmatrix} + \begin{pmatrix}
0 me te 0
0 & 0
\end{pmatrix} = \begin{pmatrix}
3 me te 7
5 & 9
\end{pmatrix} \]
3. Matū Whakarara
He matihiko tapawhā te matihiko whakarara, ko ngā huānga katoa i waho o te whakarara matua he 0. Ka rerekē pea ngā huānga i runga i te whakarara matua, engari me 0 katoa ngā huānga i waho o te whakarara matua.
Tauira:
\[ D = \begin{pmatrix}
6 me te 0 me te 0
0 me te 3 me te 0
0 & 0 & 8
\end{pmatrix} \]
Pātai:
He matihiko whakarara te matihiko e whai ake nei?
\[ C = \begin{pmatrix}
5 me te 0
0 & 6
\end{pmatrix} \]
Kōrero:
He matihiko tapawhā a C, ko ngā huānga kei waho o te hauroki matua he 0 katoa. Nō reira, he matihiko hauroki tūturu a \( C \).
4. Matū Tauine
He āhua motuhake te matihiko tauine o tētahi matihiko whakarara, ā, he ōrite ngā huānga katoa o te whakarara matua. Ka taea te whakaaro ki tētahi matihiko tauine hei whakarea tauine i runga i tētahi matihiko tuakiri.
Tauira:
\[ S = \begin{pmatrix}
4 me te 0
0 & 4
\end{pmatrix} \]
Pātai:
Whakamātauria ko te matihiko \(T\) i raro nei he matihiko tauine:
\[ T = \begin{pmatrix}
7 me te 0 me te 0
0 me te 7 me te 0
0 & 0 & 7
\end{pmatrix} \]
Kōrero:
He matihiko whakarara te matihiko \(T\) kei reira ngā huānga katoa o te whakarara matua he 7. Nō reira, he matihiko tauine a \(T\).
5. Matū Āhuarite
He matihiko tapawhā rite te matihiko taurite e ōrite ana ki tōna whakawhiti. Ko te tikanga he ōrite ngā huānga taurite huri noa i te whakarara matua, arā, \(A_{ij} = A_{ji}\) mō ia \(i\) me \(j\).
Tauira:
\[ A = \begin{pmatrix}
2 me te 1 me te 3
1 me te 4 me te 5
3 & 5 & 6
\end{pmatrix} \]
Pātai:
Tirohia mēnā he matihiko ōrite te matihiko e whai ake nei:
\[ B = \begin{pmatrix}
1 me te 2
2 & 3
\end{pmatrix} \]
Kōrero:
Ko te whakawhiti o \(B\) ko:
\[ B^T = \begin{pmatrix}
1 me te 2
2 & 3
\end{pmatrix} \]
Nā te mea ko \( B = B^T \), he matihiko ōrite a \( B \).
6. Matawhā Tapatoru
E rua ngā momo o ngā matihiko tapatoru: te tapatoru o runga me te tapatoru o raro. Ko ngā huānga katoa o raro i te matihiko tapatoru o runga he ōrite ki te 0, ko ngā huānga katoa o runga ake i te matihiko tapatoru o raro he ōrite ki te 0.
Tauira o te Tapatoru o Runga:
\[ U = \begin{pmatrix}
2 me te 3 me te 4
0 me te 5 me te 6
0 & 0 & 7
\end{pmatrix} \]
Tauira o te Tapatoru o Raro:
\[ L = \begin{pmatrix}
8 me te 0 me te 0
5 me te 6 me te 0
3 & 4 & 2
\end{pmatrix} \]
Pātai:
Whakatauhia ngā momo matihiko e whai ake nei:
\[ C = \begin{pmatrix}
1 me te 2
0 & 3
\end{pmatrix} \]
Kōrero:
Nā te mea ko ngā huānga katoa i raro i te hauroki matua he 0, ko \( C \) he matihiko tapatoru o runga.
7. Matrix Ā-Whārite
Ko te matihiko ā-whatu he matihiko tapawhā \(A\) e tutuki ana i te whārite \(A^TA = AA^T = I\), ko \(A^T\) te whakawhiti o \(A\) ā, ko \(I\) te matihiko tuakiri.
Tauira:
\[ Q = \begin{pmatrix}
1/2 me te \sqrt{3}/2 \\
\sqrt{3}/2 me te -1/2
\end{pmatrix} \]
Pātai:
Manatokohia mēnā he orthogonal ngā matihiko i raro nei:
\[ P = \begin{pmatrix}
0 me te 1
1 & 0
\end{pmatrix} \]
Kōrero:
Tuatahi ka tatauhia e tātou te whakawhiti o \(P\):
\[ P^T = \begin{pmatrix}
0 me te 1
1 & 0
\end{pmatrix} \]
Kātahi ka tatauhia e mātou \( P^TP \):
\[ P^TP = \begin{pmatrix}
0 me te 1
1 & 0
\end{pmatrix} \begin{pmatrix}
0 me te 1
1 & 0
\end{pmatrix} = \begin{pmatrix}
1 me te 0
0 & 1
\end{pmatrix} = I \]
Nā te mea ko \( P^TP = I \), he matihiko orthogonal a \(P\).
Mā te mārama ki ngā momo matihiko me ō rātou āhuatanga, ka māmā ake te whakatau i ngā otinga mō ngā raruraru pāngarau e pā ana ki ngā matihiko. He āhuatanga ahurei tō ia momo matihiko ka taea te whakamahi i roto i ngā tono pūtaiao me ngā tono hangarau.
