Namespaces
Variants
Actions

Difference between revisions of "Bilinear mapping"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex,MR,ZBL,MSC)
Line 1: Line 1:
 +
{{MSC|15}}
 +
{{TEX|done}}
 +
 
''bilinear function''
 
''bilinear function''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162801.png" /> from the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162802.png" /> of a left unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162803.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162804.png" /> and of right unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162805.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162806.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162807.png" />-[[Bimodule|bimodule]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162808.png" />, satisfying the conditions
+
A mapping $f$ from the product $V\times A$ of a left unitary $A$-module $V$ and of right unitary $B$-module $W$ into an $(A,B)$-[[Bimodule|bimodule]] $H$, satisfying the conditions
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b0162809.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628010.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628011.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628012.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628013.png" /> are arbitrarily chosen elements, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628015.png" /> are rings with a unit element. The tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628017.png" /> has the natural structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628018.png" />-bimodule. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628019.png" /> be a canonical mapping; any bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628020.png" /> will then induce a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628021.png" />-bimodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628022.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628025.png" /> is commutative, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628026.png" /> of all bilinear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628027.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628028.png" />-module with respect to the pointwise defined operations of addition and multiplication with elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628029.png" />, while the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628030.png" /> establishes a canonical isomorphism between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628031.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628032.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628033.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628034.png" /> of all linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628035.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628036.png" />.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628038.png" /> be free modules with bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628041.png" />, respectively. A bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628042.png" /> is fully determined by specifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628043.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628045.png" />, since for any finite subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628047.png" />, the following formula is valid:
+
$$f(v+v',w)=f(v,w)+f(v',w);$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$f(v,w+w')=f(v,w)+f(v,w');$$
  
Conversely, after the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628049.png" />, have been chosen arbitrarily, formula (*), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628050.png" />, defines a bilinear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628051.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628052.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628054.png" /> are finite, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628055.png" /> is said to be the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628056.png" /> with respect to the given bases.
+
$$f(av,w)=af(v,w);$$
  
Let a bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628057.png" /> be given. Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628059.png" /> are said to be orthogonal with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628060.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628061.png" />. Two subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628063.png" /> are said to be orthogonal with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628064.png" /> if any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628065.png" /> is orthogonal to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628067.png" /> is a submodule in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628068.png" />, then
+
$$f(v,wb)=f(v,w)b;$$
 +
where $v,v'\in V,\ w,w'\in W,\ a\in A,\ b\in B$ are arbitrarily chosen elements, and $A$ and $B$ are rings with a unit element. The tensor product $V\otimes W$ over $\Z$ has the natural structure of an $(A,B)$-bimodule. Let $\def\phi{\varphi}\phi:V\times W\to V\otimes W$ be a canonical mapping; any bilinear mapping $f$ will then induce a homomorphism of $(A,B)$-bimodules $\tilde f:V\otimes W\to H$ for which $f=\tilde f\circ\phi$. If $A=B$ and $A$ is commutative, then the set $L_2(V,W,H)$ of all bilinear mappings $V\times W\to H$ is an $A$-module with respect to the pointwise defined operations of addition and multiplication with elements in $A$, while the correspondence $f\mapsto \tilde f$ establishes a canonical isomorphism between the $A$-module $L_2(V,W,H)$ and the $A$-module $L(V\otimes W,H)$ of all linear mappings from $V\otimes W$ into $H$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628069.png" /></td> </tr></table>
+
Let $V$ and $W$ be free modules with bases $v_i,\ i\in I$, and $w_j$, $j\in J$, respectively. A bilinear mapping $f$ is fully determined by specifying $f(v_i,w_j)$ for all $i\in I$, $j\in J$, since for any finite subsets $I'\subset I$, $J'\subset J$, the following formula is valid:
  
which is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628070.png" />, is called the orthogonal submodule or the orthogonal complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628071.png" />. The orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628072.png" /> of the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628073.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628074.png" /> is defined in a similar way. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628075.png" /> is said to be right-degenerate (left-degenerate) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628076.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628077.png" />). The submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628079.png" /> are called, respectively, the left and right kernels of the bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628080.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628083.png" /> is said to be non-degenerate; otherwise it is said to be degenerate. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628084.png" /> is said to be a zero mapping if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628086.png" />.
+
$$f\Big(\sum_{i\in I'}a_iv_i,\sum_{j\in J'}w_jb_j\Big) = \sum_{i\in I',j\in J'} a_if(v_i,w_j)b_j.\label{1}$$
 +
Conversely, after the elements $h_{ij},\;i\in I,\;j\in J$, have been chosen arbitrarily, formula (1), where $f(v_i,w_j) = h_{ij}$, defines a bilinear mapping from $V\times W$ into $H$. If $I$ and $J$ are finite, the matrix $(f(v_i,w_j))$ is said to be the matrix of $f$ with respect to the given bases.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628087.png" />, be a set of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628088.png" />-modules, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628090.png" /> be a set of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628091.png" />-modules, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628092.png" /> be a bilinear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628093.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628094.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628095.png" /> be the direct sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628096.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628097.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628098.png" /> be the direct sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b01628099.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280100.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280101.png" />, defined by the rule
+
Let a bilinear mapping $f:V\times W\to H$ be given. Two elements $v\in V$, $w\in W$ are said to be orthogonal with respect to $f$ if $f(v,w) = 0$. Two subsets $X\subset V$ and $Y\subset W$ are said to be orthogonal with respect to $f$ if any $x\in X$ is orthogonal to any $y\in Y$. If $X$ is a submodule in $V$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280102.png" /></td> </tr></table>
+
$$X^\perp = \{w\in W:f(x,w) = 0\textrm{ for all } x\in X \},$$
 +
which is a submodule of $W$, is called the orthogonal submodule or the orthogonal complement to $X$. The orthogonal complement $Y^\perp$ of the submodule $Y$ in $W$ is defined in a similar way. The mapping $f$ is said to be right-degenerate (left-degenerate) if $V^\perp \ne \{0\}$ ($W^\perp \ne \{0\}$). The submodules $V^\perp $ and $W^\perp $ are called, respectively, the left and right kernels of the bilinear mapping $f$. If $V^\perp = \{0\}$ and $W^\perp = \{0\}$, then $f$ is said to be non-degenerate; otherwise it is said to be degenerate. The mapping $f$ is said to be a zero mapping if $V^\perp = W$ and $W^\perp = V$.
  
is a bilinear mapping and is said to be the direct sum of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280103.png" />. This is an orthogonal sum, i.e. the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280104.png" /> is orthogonal to the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280105.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280106.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280107.png" />.
+
Let $V_i,\ i\in I$, be a set of left $A$-modules, let $W_i,\ i\in I$, be a set of right $B$-modules, let $f_i$ be a bilinear mapping from $V_i\times W_i$ into $H$, let $V$ be the direct sum of the $A$-modules $V_i$, and let $W$ be the direct sum of the $B$-modules $W_i$. The mapping $f:V\times W\to H$, defined by the rule
  
The bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280108.png" /> is non-degenerate if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280109.png" /> is non-degenerate for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280110.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280111.png" /> is non-degenerate then one has
+
$$f\Big(\sum_{i\in I}v_i,\sum_{i\in I}w_i\Big) = \sum_{i\in I} f_i(v_i,w_i),$$
 +
is a bilinear mapping and is said to be the direct sum of the mappings $f_i$. This is an orthogonal sum, i.e. the submodule $V_i$ is orthogonal to the submodule $W_j$ with respect to $f$ if $i\ne j$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280112.png" /></td> </tr></table>
+
The bilinear mapping $f$ is non-degenerate if and only if $f_i$ is non-degenerate for all $i\in I$. Moreover, if $f$ is non-degenerate then one has
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016280/b016280113.png" />, a bilinear mapping is called a [[Bilinear form|bilinear form]].
+
$$V_i^\perp = \sum{j\ne i}W_j, \quad W_i^\perp = \sum{j\ne i}V_j,$$
 +
If $A=B=H$, a bilinear mapping is called a
 +
[[Bilinear form|bilinear form]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra", '''1''', Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| S. Lang,  "Algebra", Addison-Wesley  (1974)
 +
|-
 +
|}

Revision as of 23:38, 2 November 2013

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

bilinear function

A mapping $f$ from the product $V\times A$ of a left unitary $A$-module $V$ and of right unitary $B$-module $W$ into an $(A,B)$-bimodule $H$, satisfying the conditions

$$f(v+v',w)=f(v,w)+f(v',w);$$

$$f(v,w+w')=f(v,w)+f(v,w');$$

$$f(av,w)=af(v,w);$$

$$f(v,wb)=f(v,w)b;$$ where $v,v'\in V,\ w,w'\in W,\ a\in A,\ b\in B$ are arbitrarily chosen elements, and $A$ and $B$ are rings with a unit element. The tensor product $V\otimes W$ over $\Z$ has the natural structure of an $(A,B)$-bimodule. Let $\def\phi{\varphi}\phi:V\times W\to V\otimes W$ be a canonical mapping; any bilinear mapping $f$ will then induce a homomorphism of $(A,B)$-bimodules $\tilde f:V\otimes W\to H$ for which $f=\tilde f\circ\phi$. If $A=B$ and $A$ is commutative, then the set $L_2(V,W,H)$ of all bilinear mappings $V\times W\to H$ is an $A$-module with respect to the pointwise defined operations of addition and multiplication with elements in $A$, while the correspondence $f\mapsto \tilde f$ establishes a canonical isomorphism between the $A$-module $L_2(V,W,H)$ and the $A$-module $L(V\otimes W,H)$ of all linear mappings from $V\otimes W$ into $H$.

Let $V$ and $W$ be free modules with bases $v_i,\ i\in I$, and $w_j$, $j\in J$, respectively. A bilinear mapping $f$ is fully determined by specifying $f(v_i,w_j)$ for all $i\in I$, $j\in J$, since for any finite subsets $I'\subset I$, $J'\subset J$, the following formula is valid:

$$f\Big(\sum_{i\in I'}a_iv_i,\sum_{j\in J'}w_jb_j\Big) = \sum_{i\in I',j\in J'} a_if(v_i,w_j)b_j.\label{1}$$ Conversely, after the elements $h_{ij},\;i\in I,\;j\in J$, have been chosen arbitrarily, formula (1), where $f(v_i,w_j) = h_{ij}$, defines a bilinear mapping from $V\times W$ into $H$. If $I$ and $J$ are finite, the matrix $(f(v_i,w_j))$ is said to be the matrix of $f$ with respect to the given bases.

Let a bilinear mapping $f:V\times W\to H$ be given. Two elements $v\in V$, $w\in W$ are said to be orthogonal with respect to $f$ if $f(v,w) = 0$. Two subsets $X\subset V$ and $Y\subset W$ are said to be orthogonal with respect to $f$ if any $x\in X$ is orthogonal to any $y\in Y$. If $X$ is a submodule in $V$, then

$$X^\perp = \{w\in W:f(x,w) = 0\textrm{ for all } x\in X \},$$ which is a submodule of $W$, is called the orthogonal submodule or the orthogonal complement to $X$. The orthogonal complement $Y^\perp$ of the submodule $Y$ in $W$ is defined in a similar way. The mapping $f$ is said to be right-degenerate (left-degenerate) if $V^\perp \ne \{0\}$ ($W^\perp \ne \{0\}$). The submodules $V^\perp $ and $W^\perp $ are called, respectively, the left and right kernels of the bilinear mapping $f$. If $V^\perp = \{0\}$ and $W^\perp = \{0\}$, then $f$ is said to be non-degenerate; otherwise it is said to be degenerate. The mapping $f$ is said to be a zero mapping if $V^\perp = W$ and $W^\perp = V$.

Let $V_i,\ i\in I$, be a set of left $A$-modules, let $W_i,\ i\in I$, be a set of right $B$-modules, let $f_i$ be a bilinear mapping from $V_i\times W_i$ into $H$, let $V$ be the direct sum of the $A$-modules $V_i$, and let $W$ be the direct sum of the $B$-modules $W_i$. The mapping $f:V\times W\to H$, defined by the rule

$$f\Big(\sum_{i\in I}v_i,\sum_{i\in I}w_i\Big) = \sum_{i\in I} f_i(v_i,w_i),$$ is a bilinear mapping and is said to be the direct sum of the mappings $f_i$. This is an orthogonal sum, i.e. the submodule $V_i$ is orthogonal to the submodule $W_j$ with respect to $f$ if $i\ne j$.

The bilinear mapping $f$ is non-degenerate if and only if $f_i$ is non-degenerate for all $i\in I$. Moreover, if $f$ is non-degenerate then one has

$$V_i^\perp = \sum{j\ne i}W_j, \quad W_i^\perp = \sum{j\ne i}V_j,$$ If $A=B=H$, a bilinear mapping is called a bilinear form.

References

[Bo] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra", 1, Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[La] S. Lang, "Algebra", Addison-Wesley (1974)
How to Cite This Entry:
Bilinear mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_mapping&oldid=30648
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article