Transposed Matrix: Wehewehe, Nā Pono a me nā Hana

ʻO ka matrix transposed kahi manaʻo koʻikoʻi ma ke kahua o ka makemakika a me ka manaʻo matrix. Hoʻohana nui ʻia ia ma nā wahi like ʻole e like me ka ʻenekinia, physics a me ka hoʻopili ʻana, ma muli o kona hiki ke maʻalahi a hoʻoponopono i nā pilikia e pili ana i nā ʻōnaehana linear equations a me nā hoʻololi laina.

Ma mua o ka ʻimi ʻana i nā waiwai a me nā hoʻomaʻamaʻa e pili ana i ka matrix transposed, pono e hoʻomaopopo i kāna wehewehe. ʻO kahi matrix transposed kahi i loaʻa ma ka hoʻololi ʻana i nā lālani no nā kolamu o kahi matrix i hāʻawi ʻia. ʻO ia hoʻi, inā loaʻa iā mākou kahi matrix A o nā ana mxn, a laila ua hōʻike ʻia ka matrix transposed ʻo A^T a loaʻa nā ana nx m.

ʻO kekahi o nā mea kaulana loa o ka matrix transposed ʻo ia ka mālama ʻana i kekahi mau hiʻohiʻona o ka matrix kumu. No ka laʻana, inā he symmetric ka matrix A, ʻo ia hoʻi, A = A^T, a laila e mālama ʻia kēia symmetry ma kona transpose. Eia kekahi, ua like ka transpose o ka huina o na matrices me ka huina o na transposes o ia matrices.

E pili ana i ka hoʻonā ʻana, hiki i ka matrix transposed iā mākou ke maʻalahi i nā hana e like me ka hoʻonui matrix. Ma ka hoʻololi ʻana i hoʻokahi matrix a me ka hoʻonui ʻana iā ia me kekahi, loaʻa ka hopena like me ka hoʻonui ʻana i ka matrix kumu me ka hoʻololi ʻana o ka matrix lua. He waiwai nui kēia waiwai i ka hoʻoponopono ʻana i nā ʻōnaehana o nā hoʻohālikelike linear, maʻalahi i ke kaʻina hana a mālama i ka manawa.

I ka hōʻuluʻulu ʻana, ʻo ka transposed matrix kahi manaʻo koʻikoʻi i ka loiloi matrix a hāʻawi i nā pono he nui i ka hoʻoponopono ʻana i nā pilikia makemakika a me ka ʻepekema. Ma kēia ʻatikala e ʻimi hohonu mākou i nā waiwai a me nā hana e pili ana i ka matrix transposed, i hiki iā ʻoe ke hoʻohana i kēia kumuwaiwai ikaika. me ka maikaʻi i kāu mau haʻawina a me kāu mau noi.

1. Introduction to transpose matrix

ʻO ka transposed matrix kahi hana maʻamau i ka algebra linear i loaʻa nā noi like ʻole i ka ʻepekema a me ka ʻenehana. He matrix ka hopena mai ka hoʻololi ʻana i nā lālani no nā kolamu o kahi matrix kumu. He mea maikaʻi loa kēia hana, no ka mea, hiki iā mākou ke maʻalahi i ka helu ʻana a hoʻoponopono i nā pilikia e pili ana i nā ʻōnaehana hoʻohālikelike a me nā hoʻololi laina. Ma kēia ʻāpana, e ʻimi mākou i nā kikoʻī pehea e loaʻa ai ka matrix transpose o kahi matrix i hāʻawi ʻia.

No ka loaʻa ʻana o ka matrix transposed o kahi matrix, pono mākou e hahai i kēia mau ʻanuʻu:

1. E hoʻomaopopo i ka matrix kumu, hiki ke hōʻike ʻia ma ke ʻano o ka papaʻaina a i ʻole ke ʻano o nā hoohalike.
2. E hoʻololi i nā lālani a me nā kolamu o ka matrix. Hōʻike kēia i nā mea i hoʻomaka mua ʻia i nā lālani e loaʻa i loko o nā kolamu, a ʻo ia hoʻi.
3. E hoʻopaʻa i ka matrix hopena hou, ʻo ia ka transpose o ka matrix kumu.

He mea nui e hoʻomaopopo ʻaʻole e hoʻololi ka matrix transposed o ka matrix rectangular i kona mau ana, aʻo ka matrix transposed o ka matrix square e hoʻomau i ke ʻano like akā ʻo kona mau mea i hoʻololi ʻia. Eia kekahi, ua like ka matrix transposed o ka matrix transposed mua me ka matrix kumu. E ʻike kākou i kēia manawa kekahi mau laʻana ʻo ia ka mea e hōʻike maikaʻi ai i kēia mau manaʻo.

Laʻana 1: Hāʻawi ʻia ka matrix A = [2 4 1; 3 5 0], e loaa ia kakou kona matrix transpose A^T. Ma ka hoʻololi ʻana i nā lālani no nā kolamu, loaʻa iā mākou ka matrix transposed A^T = [2 3; ʻEhā. 4 5].

Laʻana 2: Hāʻawi ʻia ka matrix B = [1 2 3; 4 5 6; 7 8 9], e loaa ia kakou kona matrix transpose B^T. Ma ka hoʻololi ʻana i nā lālani no nā kolamu, loaʻa iā mākou ka matrix transposed B^T = [1 4 7; 2 5 8; 3 6 9].

I ka hōʻuluʻulu ʻana, ʻo ka transposed matrix kahi mea hana maʻamau i ka algebra linear e hiki ai iā mākou ke maʻalahi i ka helu ʻana a hoʻoponopono i nā pilikia e pili ana i nā ʻōnaehana o nā hoʻohālikelike a me nā loli linear. ʻO ka hoʻololi ʻana i nā lālani no nā kolamu o kahi matrix hiki iā mākou ke loaʻa i kāna matrix transposed, hiki ke hoʻohana ʻia ma nā ʻano hana like ʻole e like me ka physics, engineering and computing.

2. Wehewehe o ka matrix transposed

ʻO ka matrix transposed kahi matrix i loaʻa ma ka hoʻololi ʻana i nā lālani no nā kolamu ma kahi matrix i hāʻawi ʻia. He mea maikaʻi loa kēia hana i ka makemakika a me ka hoʻolālā ʻana, no ka mea e hiki ai i nā hana a me nā helu ke hoʻokō pono ʻia.

No ka loaʻa ʻana o ka matrix transposed, pono e hahai ʻia kēia mau hana:

- ʻO ka mua, ʻike ʻia ka helu o nā lālani a me nā kolamu o ka matrix kumu. He mea nui kēia e ʻike pehea e hoʻololi ʻia ai nā lālani a me nā kolamu i ka matrix hou.
- A laila, hana ʻia kahi matrix hou me ka helu o nā lālani e like me ka helu o nā kolamu o ka matrix kumu, a me ka helu o nā kolamu e like me ka helu o nā lālani o ka matrix kumu.
- A laila, hoʻololi ʻia nā lālani no nā kolamu. No ka hana ʻana i kēia, lawe ʻia ka mea ma ke kūlana i, j o ​​ka matrix kumu a kau ʻia ma ke kūlana j, i o ka matrix transposed.
- Hana hou ʻia kēia kaʻina hana no kēlā me kēia mea o ka matrix kumu, a hiki i ka pau ʻana o ka matrix transposed holoʻokoʻa.

He mea nui e hoʻomaopopo i ka matrix transposed o kahi matrix transposed ʻo ia ka matrix kumu. Hoʻohui, mālama ka matrix transposed i kekahi mau waiwai o ka matrix kumu, e like me ka hoʻohui a me ka hoʻonui. Hoʻomaʻamaʻa ka matrix transposed i ka helu ʻana i nā mea hoʻoholo, hoʻohuli, a me nā hana matrix ʻē aʻe. He mea paahana kumu ia ma ka algebra linear a ma nā wahi he nui o ka ʻepekema a me ka ʻenekinia. [HOPE

3. Heluhelu o ka matrix transposed

He hana maʻamau ia ma ka algebra linear e hoʻololi ana i nā lālani no nā kolamu o kahi matrix i hāʻawi ʻia. He mea pono loa kēia hana ma nā ʻano like ʻole e like me ka physics, engineering and computing.

No ka helu ʻana i ka matrix transpose, pono e hahai ʻia kēia mau ʻanuʻu:

• E ʻike i ka matrix mua āu e makemake ai e hoʻololi.
• E hoʻololi i nā lālani no nā kolamu, ʻo ia hoʻi, kau i nā mea o ka lālani mua e like me ke kolamu mua, nā mea o ka lālani ʻelua e like me ke kolamu ʻelua, a pēlā aku.
• ʻO ka hopena i loaʻa ʻo ia ka matrix transposed makemake ʻia.

He mea nui e hoʻomanaʻo i ka matrix transposed o kahi matrix i hoʻololi mua ʻia e like me ka matrix kumu. Eia kekahi, ua paʻa ka matrix transposed i kekahi mau waiwai nui, e like me ka huina o nā matrices i hoʻololi ʻia ua like ia me ka huina hoʻololi o nā matrices kumu.

4. Nā waiwai o ka transpose o ka matrix

ʻO ka matrix transposed kahi hana koʻikoʻi i ka algebra linear e hoʻololi ana i nā lālani no nā kolamu. Hoʻohana ʻia kēia hana ma nā kahua like ʻole, e like me ka hoʻoponopono ʻana i nā ʻōnaehana o nā kaulike laina a me ka hōʻike kiʻi kiʻi o ka ʻikepili.

No ka loaʻa ʻana o ka matrix transposed o kahi matrix i hāʻawi ʻia, pono mākou e hahai i kēia mau ʻanuʻu:

1. E ʻike i ka matrix kumu, a mākou e kuhikuhi ai ʻo A.
2. E lawe i na elemu mai ke kolamu mua o A a e kau ia lakou ma ka lalani mua o ka matrix i hoololiia, i kapaia he A^T.
3. E hana hou i ka hana mua no na kolamu a pau o A, e kau i na mea e pili ana ma na lalani o A^T.

He mea nui e hoʻomaopopo i ka matrix transposed o kahi matrix transposed ʻo ia ka matrix kumu ponoʻī, ʻo ia hoʻi (A^T)^T = A.

He mau waiwai koʻikoʻi ko ka transposed matrix e hiki ai iā mākou ke maʻalahi i ka helu ʻana a loaʻa maʻalahi nā hopena. ʻO kekahi o kēia mau waiwai:

– Ua like ka huina o na matrices i hoololiia me ka huina hoololi o na matrices kumu: (A + B)^T = A^T + B^T.
– Ua like ka hoonui scalar o ka helu maoli a me ka matrix transposed me ka hoololi ana o ka hua scalar o ia helu a me ka matrix kumu: (kA)^T = k(A^T).
– Ua like ka transpose o ka hoonui ana o na matrices elua me ka hoonui ana o na transposes ma ka hoololi hope: (AB)^T = B^TA^T.

Hāʻawi kēia mau waiwai iā mākou i nā mea hana e maʻalahi i nā hana algebraic me nā matrices transposed a loaʻa nā hopena me ka pono. He mea nui e noʻonoʻo i kēia mau waiwai a hoʻopili pono iā lākou i ka hoʻomohala ʻana i nā helu a me nā pilikia e pili ana i nā matrices a me nā ʻōnaehana o nā hoʻohālikelike laina.

5. Ka waiwai o ka transpose o ka huina o na matrices

Hoʻokumu ia i ka hoʻololi ʻana o ka huina o ʻelua matrices ua like ia me ka huina o nā transposes o ia mau matrices. 'O ia ho'i, hiki iā mākou ke loa'a ka transpose o ka huina o nā matrices ma ka ho'ohui 'ana i nā matrices a laila lawe i ka transpose o ka hopena.

No ka hōʻike ʻana i kēia waiwai, hiki iā mākou ke hoʻohana i ka wehewehe o ka transpose o kahi matrix: hoʻololi i nā lālani no nā kolamu. Ina paha he elua matrices A a me B. ʻO ka huina o kēia mau matrices ʻo A + B. A laila, lawe mākou i ka transpose o kēia huina: (A + B)^T. No ka loaʻa ʻana o ka transpose o A + B, lawe wale mākou i ka transpose o kēlā me kēia mea o ka huina.

E nānā kākou i kekahi laʻana e hoʻomaopopo pono i kēia waiwai. Inā loaʻa iā mākou nā matrices A = [1 2 3] a me B = [4 5 6]. Inā hoʻohui mākou i kēia mau matrices, loaʻa iā mākou A + B = [5 7 9]. I kēia manawa, lawe mākou i ka transpose o kēia huina: (A + B)^T = [5 7 9]^T = [5 7 9]. Hiki iā mākou ke nānā i ka hopena o ka lawe ʻana i ka transpose o ka huina like me ka huina o nā transposes o nā matrices kumu.

6. Ka waiwai o ka transpose o ka hoonui matrix

He mea paahana nui ia ma ka algebra linear. Hōʻike kēia waiwai ua like ka hoʻololi ʻana o ka huahana o nā matrices ʻelua me ka huahana o nā transposes o nā matrices pākahi akā ma ke ʻano hoʻohuli. ʻO ia hoʻi, inā he mau matrices ʻo A a me B, a laila ua like ka hoʻololi ʻana o ka huahana AB me ka transpose o B i hoʻonui ʻia me ka transpose o A.

No ka hōʻoia ʻana i kēia waiwai, e noʻonoʻo kākou i ʻelua matrices A a me B. ʻO ka mua, hoʻonui mākou i nā matrices A a me B a loaʻa ka matrix AB. A laila, helu mākou i ka transpose o ka matrix AB, i hōʻike ʻia me (AB)^T. A laila, helu mākou i ka transpose o A a me ka transpose o B, i hōʻike ʻia ʻo A^T a me B^T pakahi. ʻO ka hope, hoʻonui mākou i ka B^T me ka A^T a nānā inā ua like ka hopena me (AB)^T. Inā like nā huahana ʻelua, a laila paʻa ka waiwai.

Eia kekahi laʻana e hōʻike i ka . Inā loaʻa iā mākou nā matrices A = [[1, 2, 3], [4, 5, 6]] a me B = [[7, 8], [9, 10], [11, 12]]. E hoʻonui mua mākou i nā matrices A a me B a loaʻa ka matrix AB. A laila, helu mākou i ka transpose o AB a loaʻa ka matrix (AB)^T. A laila, helu mākou i ka transpose o A a me B, a ma kēia hihia ʻo A^T = [[1, 4], [2, 5], [3, 6]] a me B^T = [[7, 9] 11], [8, 10, 12]]. ʻO ka hope, hoʻonui mākou i ka B^T me A^T a loaʻa ka matrix B^T * A^T. Inā paʻa ka waiwai, pono e like ka hopena o B^T * A^T (AB)^T.

7. Ka waiwai o ka transpose o ka hua kiko o ka matrix

He manaʻo koʻikoʻi ia ma ke kahua o ka makemakika a me ka algebra linear. Hōʻike kēia waiwai i ka hoʻololi ʻana o ka hua kiko o nā matrices ʻelua i like me ka huina kiko o nā transposes o nā matrices i ʻōlelo ʻia. Ua wehewehe ʻia ke kaʻina hana ma lalo nei i kēlā me kēia ʻanuʻu e hoʻoponopono kēia pilikia:

1. ʻO ka mea mua, pono e hoʻomanaʻo e loaʻa ka transpose o kahi matrix ma ka hoʻololi ʻana i nā lālani no nā kolamu. No laila, inā loaʻa iā mākou nā matrices ʻelua A a me B, ua kuhikuhi ʻia nā transpose o kēia mau matrices ʻo A^T a me B^T, pakahi.

2. Ua wehewehe ʻia ka hua kiko ma waena o nā matrices ʻelua e like me ka huina o nā huahana o nā mea pili o nā matrices. ʻO ia hoʻi, inā loaʻa iā mākou ʻelua matrices A a me B o nā ana (mxn), ua helu ʻia ka hua kiko ma ka hoʻonui ʻana i nā mea o ke kūlana like a hoʻohui iā lākou.

3. No ka hooiaio ana i ka , pono e hoike ia (AB)^T = B^TA^T. Ke ulu nei ʻaoʻao ʻelua Mai ka hoʻohālikelike, hiki iā mākou ke ʻike i nā mea o ka matrix i loaʻa i nā hihia ʻelua ua like, e hōʻoia ana i ka waiwai.

Ma ka hōʻuluʻulu manaʻo, ua like ka hoʻololi ʻana o ka huahana scalar o nā matrices ʻelua me ka huahana scalar o nā transposes o ia matrices. Hāʻawi kēia manaʻo iā mākou e maʻalahi a hōʻike i nā hana makemakika like ʻole ma ke kahua o ka algebra linear. ʻO ka hoʻomanaʻo ʻana i nā wehewehe a me ka hahai ʻana i ke kaʻina hana i kēlā me kēia ʻanuʻu ke kī i ka hoʻomaopopo ʻana a me ka hoʻohana ʻana i kēia waiwai o me ka maikaʻi.

8. Nā laʻana o nā matrices i hoʻololi ʻia

No ka hoʻomaopopo maikaʻi ʻana i ka manaʻo o nā matrices transposed, pono e nānā i kekahi mau laʻana. A laila, e hōʻike ʻia ʻekolu mau laʻana e hōʻike ana i ka hana ʻana o ka matrix transposition.

Laʻana 1: E noʻonoʻo kākou i ka matrix A o ka nui 3×3:
"ʻAʻole maopopo
A = [[1, 2, 3],
[4, 5, 6],
[7, 8, 9]
"ʻAʻole maopopo
No ka loaʻa ʻana o ka matrix transposed o A, hoʻololi wale mākou i nā lālani no nā kolamu. No laila, ʻo ka matrix i hoʻololi ʻia o A, i hōʻike ʻia ʻo A^T, penei:
"ʻAʻole maopopo
A^T = [[1, 4, 7],
[2, 5, 8],
[3, 6, 9]
"ʻAʻole maopopo

Laʻana 2: Inā loaʻa iā mākou kahi matrix B o ka nui 2×4:
"ʻAʻole maopopo
B = [[1, 2, 3, 4],
[5, 6, 7, 8]
"ʻAʻole maopopo
Loaʻa ka matrix transposed o B, B^T, ma ka hoʻololi ʻana i nā lālani no nā kolamu. No laila, ʻo ka matrix i hoʻololi ʻia o B:
"ʻAʻole maopopo
B^T = [[1, 5],
[2, 6],
[3, 7],
[4, 8]
"ʻAʻole maopopo

Laʻana 3: E noʻonoʻo paha mākou he matrix C o ka nui 4×2:
"ʻAʻole maopopo
C = [[1, 2],
[3, 4],
[5, 6],
[7, 8]
"ʻAʻole maopopo
Loaʻa ka matrix transposed o C, C^T, ma ka hoʻololi ʻana i nā lālani no nā kolamu. No laila, ʻo ka matrix i hoʻololi ʻia o C:
"ʻAʻole maopopo
C^T = [[1, 3, 5, 7],
[2, 4, 6, 8]
"ʻAʻole maopopo

No laila hiki ke helu ʻia nā matrices i hoʻololi ʻia no nā nui like ʻole a me nā ʻike. ʻO ka transposition o kahi matrix kahi hana koʻikoʻi ma ke kahua o ka makemakika a hoʻohana ʻia i nā noi like ʻole, e like me ka hoʻonā ʻana i nā ʻōnaehana o ka hoʻohālikelike a me ka hoʻoponopono ʻana i ka ʻikepili i ka helu helu.

9. Pehea e hana ai i nā hana me nā matrices transposed

I ka hana ʻana me nā matrices transposed, pono e hoʻomaopopo i ka hana ʻana i nā hana maʻamau e hoʻoponopono a hoʻoponopono i nā pilikia pili iā lākou. Ma lalo iho, e hōʻike ʻia ke kaʻina hana i kēlā me kēia ʻanuʻu e hoʻokō ai i kēia mau hana:

1. Loaʻa i ka matrix transposed: No ka loaʻa ʻana o ka matrix transposed o kahi matrix i hāʻawi ʻia, pono e hoʻololi ʻia nā lālani me nā kolamu. Hoʻokō ʻia kēia ma ke kau ʻana i nā mea lālani i ke kūlana e pili ana i nā kolamu a me ka hope. Hiki ke hana me ka lima a i ʻole ka hoʻohana ʻana i nā mea hana kūikawā a i ʻole nā ​​lako polokalamu.

2. Ka huina o nā matrices i hoʻololi ʻia: Hana ʻia ka hoʻohui ʻana o ʻelua matrices transposed ma ka hoʻohui ʻana i nā mea pili i ke kūlana like o nā matrices ʻelua. He mea nui e hōʻoia i ka nui o nā matrices, ʻo ia hoʻi, loaʻa iā lākou ka helu like o nā lālani a me nā kolamu.

3. Hoʻonui ʻia matrix hoʻololi: Hana ʻia ka hoʻonui ʻana i ʻelua matrices i hoʻololi ʻia ma ka hoʻonui ʻana i kēlā me kēia mea o ka matrix transposed o ka matrix mua me ka mea pili o ka matrix transposed lua. ʻO ka hopena, he ʻano hou i loaʻa i nā ana ʻokoʻa ma mua o nā ʻāpana kumu.

10. E hoʻomaʻamaʻa me ka matrix transposed

ʻO ka matrix transposed kahi matrix i loaʻa ma ka hoʻololi ʻana i nā lālani a me nā kolamu o kahi matrix i hāʻawi ʻia. Pono kēia hana i ka algebra linear a hiki ke hoʻohana ʻia i nā matrices o kēlā me kēia nui. Aia ma lalo iho kahi pūʻulu o nā hoʻomaʻamaʻa e kōkua iā ʻoe e hoʻomaʻamaʻa me ka matrix transposed a hoʻohui i kou ʻike ma kēia kumuhana.

1. Heluhelu matrix hoololi: I haawiia i ka matrix A, e helu i kona matrix i hoololiia A^T. E hoʻomanaʻo i ka loaʻa ʻana o ka matrix transposed, pono ʻoe e hoʻololi i nā lālani no nā kolamu o A. E hoʻohana i ke kumu A._ij = A_ji e helu i nā mea o ka matrix transposed.

2. Ho'āʻo hōʻoia waiwai matrix i hoʻololi ʻia: E hōʻoia i ka like ka matrix transposed o ka matrix transposed o A me ka matrix kumu A. No ka hana ʻana i kēia, e helu mua i ka matrix transpose o A a laila ka matrix transpose o ka matrix transpose o A. E nānā inā ua like nā matrices ʻelua me ka hoʻohana ʻana i ka waiwai kaulike matrix.

11. Nā hoʻonā i nā hoʻomaʻamaʻa matrix transposed

Ma kēia ʻāpana, e ʻimi mākou i nā hoʻonā i nā hoʻomaʻamaʻa e pili ana i ka transpose matrix. Ma mua o ke komo ʻana i nā haʻawina, pono e hoʻomaopopo i ke ʻano o ka matrix transposed. ʻO kahi matrix transposed kahi i hoʻololi ʻia nā lālani no nā kolamu, ʻo ia hoʻi, lilo nā mea o ka lālani i nā mea o ke kolamu i.

No ka hoʻoponopono ʻana i nā hoʻomaʻamaʻa pili i ka matrix transposed, e hahai i kēia mau ʻanuʻu:

1. E ʻike i ka matrix i hāʻawi ʻia: E ʻike maopopo ʻoe i ka matrix āu e hana nei. Hiki i kēia matrix ke hoʻonohonoho i nā helu a i ʻole nā ​​mea hoʻololi.

2. E huli i ka matrix transposed: No ka huli ana i ka matrix transposed, pono e hoololi i na lalani no na kolamu. Hiki iā ʻoe ke hana ma ke kākau ʻana i nā mea o ka lālani mua o ka matrix kumu ma ke kolamu mua o ka matrix transposed, nā mea o ka lālani ʻelua ma ke kolamu ʻelua, a pēlā aku.

3. E nānā i ka hoʻonā: Ke loaʻa iā ʻoe ka matrix transposed, e nānā i kāu pane ma ka hōʻoia ʻana ua hoʻololi pololei ʻia nā mea. Hiki iā ʻoe ke hana i kēia ma ka hoʻohālikelike ʻana i ka matrix transposed i loaʻa me ka wehewehe ʻana o ka matrix transposed.

E hoʻomanaʻo e hoʻomaʻamaʻa me nā laʻana hou e kamaʻāina i ke kaʻina hana o ka ʻimi ʻana i ka matrix transpose. Mai hoʻokaʻulua e hoʻohana i nā mea hana e like me ka matrix calculators e nānā i kāu mau pane a hoʻomaikaʻi i kou mākaukau i ka hoʻoponopono ʻana i kēia mau hoʻomaʻamaʻa!

12. Nā noi o ka matrix transposed i ka hoʻoholo ʻana i nā ʻōnaehana o nā kaulike laina

13. Hoʻohana i ka matrix transposed i ka helu ʻana i nā mea hoʻoholo

I ka hoʻoholo ʻana i nā mea hoʻoholo matrix, hiki ke maʻalahi i ka helu ʻana me ka hoʻohana ʻana i ka matrix transposed. Loaʻa ka matrix transposed ma ka hoʻololi ʻana i nā lālani no nā kolamu o kahi matrix i hāʻawi ʻia. I kēia hihia, hiki iā mākou ke hoʻohana i ka matrix transpose e helu i nā mea hoʻoholo o nā matrices square.

ʻO ke kaʻina hana e hoʻohana ai i ka matrix transposed i ka helu ʻana i nā mea hoʻoholo penei:

• E kiʻi i ka matrix kumu āu e makemake ai e helu i ka mea hoʻoholo.
• E helu i ka matrix i hoʻololi ʻia ma ka hoʻololi ʻana i nā lālani no nā kolamu.
• E hoʻohana i ke ʻano helu helu koho (no ka laʻana, ke ʻano cofactor a i ʻole ke ʻano hoʻopau Gauss-Jordan) i ka matrix transposed.
• E lawe i ka hopena i loaʻa ma ke ʻano he mea hoʻoholo o ka matrix kumu.

Hiki iā ia ke maʻalahi i ke kaʻina hana, ʻoi aku hoʻi i ka wā e pili ana i nā make nui. Hiki ke hoʻohana ʻia kēia ʻenehana i nā noi makemakika a ʻepekema like ʻole, e like me ka hoʻonā ʻana i nā ʻōnaehana o nā hoʻohālikelike laina a i ʻole ka helu ʻana i nā wahi a me nā puke i ke geometry. E ho'āʻo e hoʻohana i ka matrix transposed i ka manawa aʻe e pono ai ʻoe e helu i kahi mea hoʻoholo a ʻike i ka maikaʻi o ia!

14. Ka hopena a me ka hōʻuluʻulu manaʻo o ka matrix transposed a me kona mau waiwai

I ka hopena, ʻo ka transposed matrix kahi hana koʻikoʻi i ka algebra linear e hiki ai iā mākou ke hoʻololi i nā lālani no nā kolamu. He mau waiwai koʻikoʻi kēia hana i pono i nā ʻano hana makemakika a me ka ʻepekema kamepiula. A laila, e hōʻuluʻulu mākou i nā waiwai kūpono o ka matrix transposed:

• Ua like ka transpose o ka matrix A me ka matrix kumu: (A^T)^T = A.
• Ua like ka transpose o ka huina o elua matrices me ka huina o na transposes o ia mau matrices: (A + B)^T = A^T + B^T.
• Ua like ka transpose o ka hua o ka matrix a me ka scalar me ka hoonui o ka scalar a me ka transpose o ka matrix: (kA)^T = k(A^T).
• Ua like ka transpose o ka hua o na matrices elua me ka hoonui o na transposes o ia mau matrices, aka, ma ke kaana hope: (AB)^T = B^TA^T.

Pono kēia mau waiwai no ka hoʻoponopono ʻana i nā matrices transposed a me ka hoʻomaʻamaʻa ʻana i nā ʻōlelo makemakika. Hoʻohana ʻia ka matrix transposed i nā noi kūpono he nui, e like me ka hoʻoponopono ʻana i nā ʻōnaehana o nā hoʻohālikelike linear, diagonalizing matrices, a me ka nānā ʻana i nā hale laina. He mea nui kona ʻike a me ka mākaukau ma ke aʻo ʻana i ka algebra linear.

I ka hōʻuluʻulu ʻana, ʻo ka transposed matrix kahi mea hana ikaika i ka linear algebra e hiki ai iā mākou ke hoʻololi i nā lālani no nā kolamu. ʻO kāna mau waiwai e hiki ai iā mākou ke hoʻomaʻamaʻa a hoʻoponopono i nā ʻōlelo makemakika me ka maikaʻi. He mea nui e hoʻomanaʻo i nā waiwai koʻikoʻi e like me ka hoʻohana ʻana i nā pōʻaiapili a me nā noi. E hoʻomaʻamaʻa a e ʻimi i nā laʻana like ʻole e hoʻomaikaʻi i kou ʻike a me kou mākaukau me nā matrices transposed.

I ka hōʻuluʻulu ʻana, ʻo ka transposed matrix kahi mea hana ikaika i ke kula o ka makemakika a me ka hoʻoponopono ʻana i nā pilikia e pili ana i nā ʻōnaehana o nā hoʻohālikelike linear. Ma ka hoʻololi wale ʻana i nā lālani i kolamu, hiki iā mākou ke loaʻa i kahi matrix transposed e hāʻawi iā mākou i ka ʻike waiwai e pili ana i nā waiwai a me nā ʻano o kahi ʻōnaehana i hāʻawi ʻia.

Ua ʻimi mākou i ka wehewehe a me nā waiwai kumu o ka matrix transposed, a ua ʻimi mākou i kekahi mau hana kūpono i hiki ai iā mākou ke hoʻomaopopo maikaʻi i kona pono a me nā noi. ma ke ao nei maoli.

He mea nui ia e hōʻike i ka transposed matrix he mea koʻikoʻi i nā ʻano ʻano like ʻole, e like me ka ʻenekinia, ʻoihana waiwai, physics a me ka ʻepekema kamepiula, a me nā mea ʻē aʻe. He mea nui kona ʻike a me ka mākaukau no ka poʻe makemake e komo hohonu i loko o kēia mau kahua a hoʻohana i ka makemakika ma ke ʻano he mea hana ikaika no ka hoʻoholo pilikia a me ka hoʻoholo ʻana.

I ka hopena, ʻo ka transposed matrix kahi mea hana makemakika waiwai nui, e hiki ai iā mākou ke hoʻopunipuni a kālailai i ka ʻikepili pono. ʻO kāna ʻike kūpono e hiki ai iā mākou ke hoʻoponopono i nā pilikia me ka maikaʻi a hoʻomohala i nā hopena hou i nā ʻano like ʻole.