Carte multilinéaire

En algèbre linéaire , une application multilinéaire est une fonction de plusieurs variables qui est linéaire séparément par rapport à chaque variable. Plus précisément, une application multilinéaire est une fonction

{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{, <semantics><mrow><mstyle><mi>f</mi><mo>:</mo><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>W</mi><mrow><mtext>,</mtext></mrow></mstyle></mrow><annotation>{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{,}}}</annotation></semantics></math></span><img src=

où ${\displaystyle V_{1},\ldots ,V_{n <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub></mstyle></mrow><annotation>{ displaystyle V_ {1}, ldots, V_ {n}}</annotation></semantics></math><img src=$ ${\displaystyle n\in \mathbb {Z} _{\geq 0 <semantics><mrow><mstyle><mi>n</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle n\in \mathbb {Z} _{\geq 0}}</annotation></semantics></math><img src=$ ${\displaystyle W <semantics><mrow><mstyle><mi>W</mi></mstyle></mrow><annotation>{\displaystyle W}</annotation></semantics></math><img src=$ ${\displaystyle i <semantics><mrow><mstyle><mi>je</mi></mstyle></mrow><annotation>{\displaystyle i}</annotation></semantics></math><img src=$ ${\displaystyle v_{i <semantics><mrow><mstyle><msub><mi>v</mi><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle v_{i}}</annotation></semantics></math><img src=$ ${\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n}) <semantics><mrow><mstyle><mi>f</mi><mo>(</mo><msub><mi>v</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>v</mi><mrow><mi>je</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>v</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}</annotation></semantics></math><img src=$ ${\displaystyle v_{i <semantics><mrow><mstyle><msub><mi>v</mi><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle v_{i}}</annotation></semantics></math><img src=$ ${\displaystyle 2^{2 <semantics><mrow><mstyle><msup><mn>2</mn><mrow><mn>2</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle 2^{2}}</annotation></semantics></math><img src=$

Une application multilinéaire d'une variable est une application linéaire , et une application multilinéaire de deux variables est une application bilinéaire . Plus généralement, pour tout entier non négatif ${\displaystyle k <semantics><mrow><mstyle><mi>k</mi></mstyle></mrow><annotation>{\displaystyle k}</annotation></semantics></math><img src=$ k variables est appelée application k -linéaire . Si le codomaine d'une application multilinéaire est le corps des scalaires , on parle de forme multilinéaire . Les applications multilinéaires et les formes multilinéaires sont des objets d'étude fondamentaux en algèbre multilinéaire .

Si toutes les variables appartiennent au même espace, on peut considérer les applications k- linéaires symétriques , antisymétriques et alternées . Les deux dernières coïncident si l' anneau (ou le corps ) sous-jacent possède une caractéristique différente de deux ; sinon, ce sont les deux premières qui coïncident.

Exemples

Toute application bilinéaire est une application multilinéaire. Par exemple, tout produit scalaire sur une application bilinéaire est une application multilinéaire. ${\displaystyle \mathbb {R} <semantics><mrow><mstyle><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbb {R} ^{3 <semantics><mrow><mstyle><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbb {R} ^{3}}</annotation></semantics></math><img src=$
Le déterminant d'une matrice carrée est une fonction multilinéaire des colonnes (ou des lignes) ; c'est également une fonction alternée des colonnes (ou des lignes).
Si ${\displaystyle F\colon \mathbb {R} ^{m} o \mathbb {R} ^{n <semantics><mrow><mstyle><mi>F</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle F\colon \mathbb {R} ^{m} o \mathbb {R} ^{n}}</annotation></semantics></math><img src=$ C ^k , alors la ${\displaystyle k <semantics><mrow><mstyle><mi>k</mi></mstyle></mrow><annotation>{\displaystyle k}</annotation></semantics></math><img src=$ ${\displaystyle F <semantics><mrow><mstyle><mi>F</mi></mstyle></mrow><annotation>{\displaystyle F}</annotation></semantics></math><img src=$ ${\displaystyle p <semantics><mrow><mstyle><mi>p</mi></mstyle></mrow><annotation>{\displaystyle p}</annotation></semantics></math><img src=$ ${\displaystyle k <semantics><mrow><mstyle><mi>k</mi></mstyle></mrow><annotation>{\displaystyle k}</annotation></semantics></math><img src=$ ${\displaystyle D^{k}\!F\colon \mathbb {R} ^{m} imes \cdots imes \mathbb {R} ^{m} o \mathbb {R} ^{n <semantics><mrow><mstyle><msup><mi>D</mi><mrow><mi>k</mi></mrow></msup><mspace></mspace><mi>F</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><mo>⋯</mo><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle D^{k}\!F\colon \mathbb {R} ^{m} imes \cdots imes \mathbb {R} ^{m} o \mathbb {R} ^{n}}</annotation></semantics></math><img src=$

Représentation des coordonnées

Laisser

{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{, <semantics><mrow><mstyle><mi>f</mi><mo>:</mo><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>W</mi><mrow><mtext>,</mtext></mrow></mstyle></mrow><annotation>{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{,}}}</annotation></semantics></math></span><img src=

soit une application multilinéaire entre espaces vectoriels de dimension finie , où ${\displaystyle V_{i}\! <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mi>je</mi></mrow></msub><mspace></mspace></mstyle></mrow><annotation>{\displaystyle V_{i}\!}</annotation></semantics></math><img src=$ ${\displaystyle d_{i}\! <semantics><mrow><mstyle><msub><mi>d</mi><mrow><mi>je</mi></mrow></msub><mspace></mspace></mstyle></mrow><annotation>{\displaystyle d_{i}\!}</annotation></semantics></math><img src=$ ${\displaystyle W\! <semantics><mrow><mstyle><mi>W</mi><mspace></mspace></mstyle></mrow><annotation>{\displaystyle W\!}</annotation></semantics></math><img src=$ ${\displaystyle d\! <semantics><mrow><mstyle><mi>d</mi><mspace></mspace></mstyle></mrow><annotation>{\displaystyle d\!}</annotation></semantics></math><img src=$ ${\displaystyle \{{ extbf {e}}_{i1},\ldots ,{ extbf {e}}_{id_{i}}\ <semantics><mrow><mstyle><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><msub><mi>d</mi><mrow><mi>je</mi></mrow></msub></mrow></msub><mo>}</mo></mstyle></mrow><annotation>{\displaystyle \{{ extbf {e}}_{i1},\ldots,{ extbf {e}}_{id_{i}}\}}</annotation></semantics></math><img src=$ ${\displaystyle V_{i}\! <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mi>je</mi></mrow></msub><mspace></mspace></mstyle></mrow><annotation>{\displaystyle V_{i}\!}</annotation></semantics></math><img src=$ ${\displaystyle \{{ extbf {b}}_{1},\ldots ,{ extbf {b}}_{d}\ <semantics><mrow><mstyle><mo>{</mo><msub><mrow><mrow><mtext>b</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mtext>b</mtext></mrow></mrow><mrow><mi>d</mi></mrow></msub><mo>}</mo></mstyle></mrow><annotation>{\displaystyle \{{ extbf {b}}_{1},\ldots ,{ extbf {b}}_{d}\}}</annotation></semantics></math><img src=$ ${\displaystyle W\! <semantics><mrow><mstyle><mi>W</mi><mspace></mspace></mstyle></mrow><annotation>{\displaystyle W\!}</annotation></semantics></math><img src=$ ${\displaystyle A_{j_{1}\cdots j_{n}}^{k <semantics><mrow><mstyle><msubsup><mi>UN</mi><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>k</mi></mrow></msubsup></mstyle></mrow><annotation>{\displaystyle A_{j_{1}\cdots j_{n}}^{k}}</annotation></semantics></math><img src=$

{\displaystyle f({ extbf {e}}_{1j_{1}},\ldots ,{ extbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{ extbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{ extbf {b}}_{d}. <semantics><mrow><mstyle><mi>f</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>n</mi><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mi>UN</mi><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow><mrow><mn>1</mn></mrow></msubsup><mspace></mspace><msub><mrow><mrow><mtext>b</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>UN</mi><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup><mspace></mspace><msub><mrow><mrow><mtext>b</mtext></mrow></mrow><mrow><mi>d</mi></mrow></msub><mo>.</mo></mstyle></mrow><annotation>{\displaystyle f({ extbf {e}}_{1j_{1}},\ldots ,{ extbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{ extbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{ extbf {b}}_{d}.}</annotation></semantics></math></span><img src=

Ensuite les scalaires ${\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\ <semantics><mrow><mstyle><mo>{</mo><msubsup><mi>UN</mi><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>k</mi></mrow></msubsup><mo>∣</mo><mn>1</mn><mo>≤</mo><msub><mi>j</mi><mrow><mi>je</mi></mrow></msub><mo>≤</mo><msub><mi>d</mi><mrow><mi>je</mi></mrow></msub><mo>,</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi><mo>}</mo></mstyle></mrow><annotation>{\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}}</annotation></semantics></math><img src=$ ${\displaystyle f\! <semantics><mrow><mstyle><mi>f</mi><mspace></mspace></mstyle></mrow><annotation>{\displaystyle f\!}</annotation></semantics></math><img src=$

{\displaystyle { extbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{ extbf {e}}_{ij}\! <semantics><mrow><mstyle><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mi>je</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>d</mi><mrow><mi>je</mi></mrow></msub></mrow></munderover><msub><mi>v</mi><mrow><mi>je</mi><mi>j</mi></mrow></msub><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><mi>j</mi></mrow></msub><mspace></mspace></mstyle></mrow><annotation>{\displaystyle { extbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{ extbf {e}}_{ij}\!}</annotation></semantics></math></span><img src=

pour ${\displaystyle 1\leq i\leq n\! <semantics><mrow><mstyle><mn>1</mn><mo>≤</mo><mi>je</mi><mo>≤</mo><mi>n</mi><mspace></mspace></mstyle></mrow><annotation>{\displaystyle 1\leq i\leq n\!}</annotation></semantics></math><img src=$

{\displaystyle f({ extbf {v}}_{1},\ldots ,{ extbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{ extbf {b}}_{k}. <semantics><mrow><mstyle><mi>f</mi><mo>(</mo><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>d</mi><mrow><mn>1</mn></mrow></msub></mrow></munderover><mo>⋯</mo><munderover><mo>∑</mo><mrow><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>d</mi><mrow><mi>n</mi></mrow></msub></mrow></munderover><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></munderover><msubsup><mi>UN</mi><mrow><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mi>v</mi><mrow><mn>1</mn><msub><mi>j</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>⋯</mo><msub><mi>v</mi><mrow><mi>n</mi><msub><mi>j</mi><mrow><mi>n</mi></mrow></msub></mrow></msub><msub><mrow><mrow><mtext>b</mtext></mrow></mrow><mrow><mi>k</mi></mrow></msub><mo>.</mo></mstyle></mrow><annotation>{\displaystyle f({ extbf {v}}_{1},\ldots ,{ extbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{ extbf {b}}_{k}.}</annotation></semantics></math></span><img src=

Exemple

Prenons une fonction trilinéaire

{\displaystyle g\colon R^{2} imes R^{2} imes R^{2} o R, <semantics><mrow><mstyle><mi>g</mi><mo>:</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>R</mi><mo>,</mo></mstyle></mrow><annotation>{\displaystyle g\colon R^{2} imes R^{2} imes R^{2} o R,}</annotation></semantics></math></span><img src=

où , et .

Une base pour chaque est ${\displaystyle \{{ extbf {e}}_{i1},\ldots ,{ extbf {e}}_{id_{i}}\}=\{{ extbf {e}}_{1},{ extbf {e}}_{2}\}=\{(1,0),(0,1)\}. <semantics><mrow><mstyle><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><msub><mi>d</mi><mrow><mi>je</mi></mrow></msub></mrow></msub><mo>}</mo><mo>=</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>=</mo><mo>{</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle \{{ extbf {e}}_{i1},\ldots ,{ extbf {e}}_{id_{i}}\}=\{{ extbf {e}}_{1},{ extbf {e}}_{2}\}=\{(1,0),(0,1)\}.}</annotation></semantics></math><img src=$

{\displaystyle g({ extbf {e}}_{1i},{ extbf {e}}_{2j},{ extbf {e}}_{3k})=f({ extbf {e}}_{i},{ extbf {e}}_{j},{ extbf {e}}_{k})=A_{ijk}, <semantics><mrow><mstyle><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn><mi>je</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>3</mn><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><msub><mi>UN</mi><mrow><mi>je</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo></mstyle></mrow><annotation>{\displaystyle g({ extbf {e}}_{1i},{ extbf {e}}_{2j},{ extbf {e}}_{3k})=f({ extbf {e}}_{i},{ extbf {e}}_{j},{ extbf {e}}_{k})=A_{ijk},}</annotation></semantics></math></span><img src=

où ${\displaystyle i,j,k\in \{1,2\ <semantics><mrow><mstyle><mi>je</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mstyle></mrow><annotation>{\displaystyle i,j,k\in \{1,2\}}</annotation></semantics></math><img src=$ ${\displaystyle A_{ijk <semantics><mrow><mstyle><msub><mi>UN</mi><mrow><mi>je</mi><mi>j</mi><mi>k</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle A_{ijk}}</annotation></semantics></math><img src=$ ${\displaystyle V_{i <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle V_{i}}</annotation></semantics></math><img src=$

{\displaystyle \{{ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{1}\},\{{ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{2}\},\{{ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{1}\},\{{ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{2}\},\{{ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{1}\},\{{ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{2}\},\{{ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{1}\},\{{ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{2}\}. <semantics><mrow><mstyle><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>,</mo><mo>{</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle \{{ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{1}\},\{{ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{2}\},\{{ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{1}\},\{{ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{2}\},\{{ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{1}\},\{{ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{2}\},\{{ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{1}\},\{{ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{2}\}.}</annotation></semantics></math></span><img src=

Chaque vecteur ${\displaystyle { extbf {v}}_{i}\in V_{i}=R^{2 <semantics><mrow><mstyle><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msub><mi>V</mi><mrow><mi>je</mi></mrow></msub><mo>=</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle { extbf {v}}_{i}\in V_{i}=R^{2}}</annotation></semantics></math><img src=$

{\displaystyle { extbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{ extbf {e}}_{ij}=v_{i1} imes { extbf {e}}_{1}+v_{i2} imes { extbf {e}}_{2}=v_{i1} imes (1,0)+v_{i2} imes (0,1). <semantics><mrow><mstyle><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mi>je</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><msub><mi>v</mi><mrow><mi>je</mi><mi>j</mi></mrow></msub><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mi>je</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>v</mi><mrow><mi>je</mi><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>v</mi><mrow><mi>je</mi><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>v</mi><mrow><mi>je</mi><mn>1</mn></mrow></msub><mo>×</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>+</mo><msub><mi>v</mi><mrow><mi>je</mi><mn>2</mn></mrow></msub><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle { extbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{ extbf {e}}_{ij}=v_{i1} imes { extbf {e}}_{1}+v_{i2} imes { extbf {e}}_{2}=v_{i1} imes (1,0)+v_{i2} imes (0,1).}</annotation></semantics></math></span><img src=

La valeur de la fonction pour un ensemble arbitraire de trois vecteurs ${\displaystyle { extbf {v}}_{i}\in R^{2 <semantics><mrow><mstyle><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle { extbf {v}}_{i}\in R^{2}}</annotation></semantics></math><img src=$

{\displaystyle g({ extbf {v}}_{1},{ extbf {v}}_{2},{ extbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k}, <semantics><mrow><mstyle><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>v</mtext></mrow></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>je</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><msub><mi>UN</mi><mrow><mi>je</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>v</mi><mrow><mn>1</mn><mi>je</mi></mrow></msub><msub><mi>v</mi><mrow><mn>2</mn><mi>j</mi></mrow></msub><msub><mi>v</mi><mrow><mn>3</mn><mi>k</mi></mrow></msub><mo>,</mo></mstyle></mrow><annotation>{\displaystyle g({ extbf {v}}_{1},{ extbf {v}}_{2},{ extbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}</annotation></semantics></math></span><img src=

ou sous forme développée comme

{\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace imes g({ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{1})+acf imes g({ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{2})\\&+ade imes g({ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{1})+adf imes g({ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{2})+bce imes g({ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{1})+bcf imes g({ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{2})\\&+bde imes g({ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{1})+bdf imes g({ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{2}).\end{aligned <semantics><mrow><mstyle><mrow><mtable><mtr><mtd><mi>g</mi><mo>(</mo><mo>(</mo><mi>un</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>)</mo></mtd><mtd><mi></mi><mo>,</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>un</mi><mi>c</mi><mi>e</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>un</mi><mi>c</mi><mi>f</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mi>un</mi><mi>d</mi><mi>e</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>un</mi><mi>d</mi><mi>f</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>b</mi><mi>c</mi><mi>e</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mi>b</mi><mi>d</mi><mi>e</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>×</mo><mi>g</mi><mo>(</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mtext>e</mtext></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mstyle></mrow><annotation>{\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace imes g({ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{1})+acf imes g({ extbf {e}}_{1},{ extbf {e}}_{1},{ extbf {e}}_{2})\\&+ade imes g({ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{1})+adf imes g({ extbf {e}}_{1},{ extbf {e}}_{2},{ extbf {e}}_{2})+bce imes g({ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{1})+bcf imes g({ extbf {e}}_{2},{ extbf {e}}_{1},{ extbf {e}}_{2})\\&+bde imes g({ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{1})+bdf imes g({ extbf {e}}_{2},{ extbf {e}}_{2},{ extbf {e}}_{2}).\end{aligned}}}</annotation></semantics></math></span><img src=

Relation avec les produits tensoriels

Il existe une correspondance biunivoque naturelle entre les applications multilinéaires.

{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{, <semantics><mrow><mstyle><mi>f</mi><mo>:</mo><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>W</mi><mrow><mtext>,</mtext></mrow></mstyle></mrow><annotation>{\displaystyle f\colon V_{1} imes \cdots imes V_{n} o W{ ext{,}}}</annotation></semantics></math></span><img src=

et les cartes linéaires

{\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n} o W{ ext{, <semantics><mrow><mstyle><mi>F</mi><mo>:</mo><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>W</mi><mrow><mtext>,</mtext></mrow></mstyle></mrow><annotation>{\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n} o W{ ext{,}}}</annotation></semantics></math></span><img src=

où ${\displaystyle V_{1}\otimes \cdots \otimes V_{n}\! <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub><mspace></mspace></mstyle></mrow><annotation>{\displaystyle V_{1}\otimes \cdots \otimes V_{n}\!}</annotation></semantics></math><img src=$ ${\displaystyle V_{1},\ldots ,V_{n <semantics><mrow><mstyle><msub><mi>V</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>V</mi><mrow><mi>n</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle V_{1},\ldots ,V_{n}}</annotation></semantics></math><img src=$ ${\displaystyle f <semantics><mrow><mstyle><mi>f</mi></mstyle></mrow><annotation>{\displaystyle f}</annotation></semantics></math><img src=$ ${\displaystyle F <semantics><mrow><mstyle><mi>F</mi></mstyle></mrow><annotation>{\displaystyle F}</annotation></semantics></math><img src=$

{\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}). <semantics><mrow><mstyle><mi>f</mi><mo>(</mo><msub><mi>v</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>v</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><msub><mi>v</mi><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><msub><mi>v</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).}</annotation></semantics></math></span><img src=

Fonctions multilinéaires sur des matrices n × n

On peut considérer les fonctions multilinéaires, sur une matrice n d'un anneau commutatif une telle matrice et , ses lignes Alors la fonction multilinéaire peut s'écrire comme

{\displaystyle D(A)=D(a_{1},\ldots ,a_{n}), <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>UN</mi><mo>)</mo><mo>=</mo><mi>D</mi><mo>(</mo><msub><mi>un</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo></mstyle></mrow><annotation>{\displaystyle D(A)=D(a_{1},\ldots ,a_{n}),}</annotation></semantics></math></span><img src=

satisfaisant

{\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}). <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><msub><mi>un</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>c</mi><msub><mi>un</mi><mrow><mi>je</mi></mrow></msub><mo>+</mo><msubsup><mi>un</mi><mrow><mi>je</mi></mrow><mo>′</mo></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>c</mi><mi>D</mi><mo>(</mo><msub><mi>un</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>je</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>+</mo><mi>D</mi><mo>(</mo><msub><mi>un</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mi>un</mi><mrow><mi>je</mi></mrow><mo>′</mo></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}</annotation></semantics></math></span><img src=

Si nous laissons faire ${\displaystyle {\hat {e}}_{j <semantics><mrow><mstyle><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mi>j</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle {\hat {e}}_{j}}</annotation></semantics></math><img src=$

{\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}. <semantics><mrow><mstyle><msub><mi>un</mi><mrow><mi>je</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>UN</mi><mo>(</mo><mi>je</mi><mo>,</mo><mi>j</mi><mo>)</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mi>j</mi></mrow></msub><mo>.</mo></mstyle></mrow><annotation>{\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.}</annotation></semantics></math></span><img src=

En utilisant la multilinéarité de nous réécrivons comme

{\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n} ight)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}). <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>UN</mi><mo>)</mo><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>UN</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>)</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>un</mi><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>UN</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>)</mo><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>un</mi><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>un</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n} ight)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}</annotation></semantics></math></span><img src=

Continuing this substitution for each we get, for ,

{\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}). <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>n</mi></mrow></munder><mo>…</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><msub><mi>k</mi><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>n</mi></mrow></munder><mo>…</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><msub><mi>k</mi><mrow><mi>n</mi></mrow></msub><mo>≤</mo><mi>n</mi></mrow></munder><mi>A</mi><mo>(</mo><mn>1</mn><mo>,</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>A</mi><mo>(</mo><mn>2</mn><mo>,</mo><msub><mi>k</mi><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>…</mo><mi>A</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mi>k</mi><mrow><mi>n</mi></mrow></msub><mo>)</mo><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><msub><mi>k</mi><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>)</mo><mo>.</mo></mstyle></mrow><annotation>{\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}</annotation></semantics></math></span><img src=

Therefore, is uniquely determined by how ${\displaystyle {\hat {e}}_{k_{1}},\dots,{\hat {e}}_{k_{n </span referrerpolicy=$ .

Example

In the case of 2×2 matrices, we get

{\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\, <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace></mstyle></mrow><annotation>{\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}</annotation></semantics></math></span><img src=

where ${\displaystyle {\hat {e}}_{1}=[1,0] <semantics><mrow><mstyle><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>]</mo></mstyle></mrow><annotation>{\displaystyle {\hat {e}}_{1}=[1,0]}</annotation></semantics></math><img src=$ ${\displaystyle {\hat {e}}_{2}=[0,1] <semantics><mrow><mstyle><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mstyle></mrow><annotation>{\displaystyle {\hat {e}}_{2}=[0,1]}</annotation></semantics></math><img src=$ ${\displaystyle D <semantics><mrow><mstyle><mi>D</mi></mstyle></mrow><annotation>{\displaystyle D}</annotation></semantics></math><img src=$ ${\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0 <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation>{\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}</annotation></semantics></math><img src=$ ${\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I) <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>−</mo><mi>D</mi><mo>(</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>e</mi><mo>^</mo></mover></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>−</mo><mi>D</mi><mo>(</mo><mi>I</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}</annotation></semantics></math><img src=$ ${\displaystyle D(I)=1 <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation>{\displaystyle D(I)=1}</annotation></semantics></math><img src=$ ×2 matrices:

{\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}. <semantics><mrow><mstyle><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>.</mo></mstyle></mrow><annotation>{\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.}</annotation></semantics></math></span><img src=

Properties

A multilinear map has a value of zero whenever one of its arguments is zero.