Translation for "permutaatiomatriisi" to english

Permutaatiomatriisi

• permutation
• permutation matrix

Translation examples

permutation

Matematiikan matriisiteoriassa permutaatiomatriisi on neliöllinen yksikkömatriisi, jolla on täsmälleen yksi johtava alkio 1 jokaisella rivillä ja sarakkeessa ja kaikialla muualla 0.
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere.

PErmutaatiomatriisi Pπ vastaa permutaatiota : π = ( 1 2 3 4 5 1 4 2 5 3 ) , {\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}},} joten P π = = = . {\displaystyle P_{\pi }={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\\\mathbf {e} _{4}\\\mathbf {e} _{2}\\\mathbf {e} _{5}\\\mathbf {e} _{3}\end{bmatrix}}={\begin{bmatrix}1&0&0&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&1&0&0\end{bmatrix}}.} Annetulle vektorille g, P π g = = . {\displaystyle P_{\pi }\mathbf {g} ={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}{\begin{bmatrix}g_{1}\\g_{2}\\g_{3}\\g_{4}\\g_{5}\end{bmatrix}}={\begin{bmatrix}g_{1}\\g_{4}\\g_{2}\\g_{5}\\g_{3}\end{bmatrix}}.} Permutaatiomatriisi on aina muotoa {\displaystyle {\begin{bmatrix}\mathbf {e} _{a_{1}}\\\mathbf {e} _{a_{2}}\\\vdots \\\mathbf {e} _{a_{j}}\\\end{bmatrix}}} jossa eai esittää i:ttä basis alkeisvektoria (kuten riviä) R:lle 'j, missä {\displaystyle {\begin{bmatrix}1&2&\ldots &j\\a_{1}&a_{2}&\ldots &a_{j}\end{bmatrix}}} on permutaatio muoto permutaatiomatriisista.
The permutation matrix Pπ corresponding to the permutation : π = ( 1 2 3 4 5 1 4 2 5 3 ) , {\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}},} is P π = = = . {\displaystyle P_{\pi }={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\\\mathbf {e} _{4}\\\mathbf {e} _{2}\\\mathbf {e} _{5}\\\mathbf {e} _{3}\end{bmatrix}}={\begin{bmatrix}1&0&0&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&1&0&0\end{bmatrix}}.} Given a vector g, P π g = = . {\displaystyle P_{\pi }\mathbf {g} ={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}{\begin{bmatrix}g_{1}\\g_{2}\\g_{3}\\g_{4}\\g_{5}\end{bmatrix}}={\begin{bmatrix}g_{1}\\g_{4}\\g_{2}\\g_{5}\\g_{3}\end{bmatrix}}.} A permutation matrix will always be in the form {\displaystyle {\begin{bmatrix}\mathbf {e} _{a_{1}}\\\mathbf {e} _{a_{2}}\\\vdots \\\mathbf {e} _{a_{j}}\\\end{bmatrix}}} where eai represents the ith basis vector (as a row) for Rj, and where {\displaystyle {\begin{bmatrix}1&2&\ldots &j\\a_{1}&a_{2}&\ldots &a_{j}\end{bmatrix}}} is the permutation form of the permutation matrix.