Iz Wikipedije, proste enciklopedije
Dualni kvaternion je v teoriji kolobarjev sestavni del nekomutativnega in neasociativnega kolobarja. Dualni kvaternioni so zgrajeni na podoben način kot običajni kvaternioni. Od njih se razlikujejo samo v tem, da jih namesto realnih števil kot koeficienti sestavljajo dualna števila.
Dualni kvaternion lahko prikažemo v obliki
![{\displaystyle q=q_{0}+\epsilon q_{\epsilon }\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad2cca69605184bf9135135516fe0e69d52fbb6)
kjer je
običajni kvaternion
dualna enota za katero velja
(nilpotentnost).
Operacije z dualnimi kvaternioni[uredi | uredi kodo]
Seštevanje dualnih kvaternionov je enostavno seštevanje njegovih koeficientov.
Dualne kvaternione množimo tako, da množimo njegove komponente.
Imamo dva dualna kvaterniona:
![{\displaystyle Q_{1}=r_{1}+\varepsilon d_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24641e692f906a9be7a40b85598ca8fc74a440e3)
in
.
Njun zmnožek je enak:
.
Pri tem pa ne nastopa
, ker je
.
To nam da naslednjo tabelo za množenje:
![{\displaystyle Q_{1}*Q_{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33d76c3d76a19888ef8f1a150f32939ca41b31f4) |
![{\displaystyle Q_{2}.1\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc07d11903d749d3be261ac81df0b1d8eb9d7d9) |
![{\displaystyle Q_{2}.i\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf468ec810b2b370270220fad3a92892cc92e9b) |
![{\displaystyle Q_{2}.j\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1189b0c35cd989eee48644cbbc7acba0bdc6ff81) |
![{\displaystyle Q_{2}.k\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd7a7f14f5b1a51634a00175a2eaaef70146d6d) |
![{\displaystyle Q_{2}.\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9f9e5b94e1d22d8dd4cb75c57d8081a8cd65c3) |
![{\displaystyle Q_{2}.\varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b02ae0ea933ef374fbec4559f4f71e05ca868a2) |
![{\displaystyle Q_{2}.\varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d452020059b83c134269114eb3526e00a27d26) |
![{\displaystyle Q_{2}.\varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9323f281063c1fb1e3171e22702ece8788d7737c) |
![{\displaystyle Q_{1}.1\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2652c0666b2d58bc92ca285d2dcc0e26bdc7e3c9) |
1 |
i |
j |
k |
![{\displaystyle \varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173) |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
![{\displaystyle Q_{1}.i\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4389d5f766783c85826e7824bcf704c4fd01338) |
i |
-1 |
k |
-j |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
![{\displaystyle -\varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a740f1647b5dca11cb0659e831776fd1f2b09ee) |
![{\displaystyle Q_{1}.j\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8519bd5ebb7ddfb95a6de10730e6b641224fcd46) |
j |
-k |
-1 |
i |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle -\varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0b9e3ad59b64e35d2476a356d7b9b2efb034b3) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
![{\displaystyle Q_{1}.k\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc67455f952b47547cee3f39952d5d362198551) |
k |
j |
-i |
-1 |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle -\varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3032f513c0da9931e07ab520c9624ef4c74671) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
![{\displaystyle Q_{1}.\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd701a49719666a15fbe550a239d90f66a938fde) |
![{\displaystyle \varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173) |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
0 |
0 |
0 |
0 |
![{\displaystyle Q_{1}.\varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a193ac84888f6a1a898e2bfae9effee15adf65) |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
![{\displaystyle -\varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a740f1647b5dca11cb0659e831776fd1f2b09ee) |
0 |
0 |
0 |
0 |
![{\displaystyle Q_{1}.\varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b5ec840554988c78e895b54394aa2368c9e3b1) |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle -\varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0b9e3ad59b64e35d2476a356d7b9b2efb034b3) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
![{\displaystyle \varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c3ce5e0706d1bffcadae232313ee52138e808e) |
0 |
0 |
0 |
0 |
![{\displaystyle Q_{1}.\varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b700f3cce64721df54a9b2d6f15af17ce0233ad1) |
![{\displaystyle \varepsilon k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b294f490a3389296854f49a36d485059df96d30) |
![{\displaystyle \varepsilon j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98be059cf146eeffa5801cc008767815edede2) |
![{\displaystyle -\varepsilon i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3032f513c0da9931e07ab520c9624ef4c74671) |
![{\displaystyle -\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a07b913c3b445bab7efbcbf683b09d53b625674) |
0 |
0 |
0 |
0 |
.
Konjugirani dualni kvaternioni[uredi | uredi kodo]
Dualni kvaternioni imajo tri konjugirane oblike:
![{\displaystyle q^{\dagger }=r^{*}+\varepsilon d^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d123d93464c0956fb96a0430bf0a69a534121b)
![{\displaystyle q_{\varepsilon }=r-\varepsilon d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdc902869e5b07a8be3fdbc1591d4295dbc0d67)
![{\displaystyle q_{\varepsilon }^{\dagger }=r^{*}-\varepsilon d^{*}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d72cdfb09c6ce1847bf8ee4c98088098ec5565)
kjer je
dualni kvaternion
realni del kvaterniona
dualni del
Obratna vrednost dualnega kvaterniona[uredi | uredi kodo]
Podobno kot pri običajnem kvaternionu, se obratna vrednost izračuna po obrazcu:
.
- kjer je
- z
označen dualni kvaternion.
Norma dualnega kvaterniona[uredi | uredi kodo]