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Chapter 5 Bilinearity
We give an introduction to a general theory of “multiplication” of vectors.
5.1 Bilinear maps
5.1.1 Definitions and examples
-
Definition. Let \(U,V,W\) be vector spaces over a field \(\F \). A map \(B:U\times V\to W\) is bilinear if it is linear in each slot separately:
\(\seteqnumber{0}{5.}{0}\)
\begin{align*}
B(\lambda u_1+u_2,v)&=\lambda B(u_1,v)+B(u_2,v)\\ B(u,\lambda v_1+v_2)&=\lambda B(u,v_1)+B(u,v_2),
\end{align*}
for all \(u,u_1,u_2\in U\), \(v,v_1,v_2\in V\) and \(\lambda \in \F \).
A bilinear map \(U\times V\to \F \) is called a bilinear pairing.
A bilinear map \(V\times V\to \F \) is called a bilinear form on \(V\).
-
. A bilinear map \(B:U\times V\to W\) has \(B(u,0)=B(0,v)=0\), for all \(u\in U\) and \(v\in V\). Indeed,
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B(u,0)=B(u,0+0)=B(u,0)+B(u,0)
\end{equation*}
and similarly for \(B(0,v)\).
-
Examples.
-
-
(1) Matrix multiplication is bilinear:
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
(A,B)\mapsto AB:M_{m\times n}(\F )\times M_{n\times k}(\F )\to M_{m\times k}(\F ).
\end{equation*}
-
(2) Composition of maps is bilinear:
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
(\psi ,\phi )\mapsto \psi \circ \phi : L(U,W)\times L(V,U)\to L(V,W).
\end{equation*}
-
(3) Evaluation \((\alpha ,v)\mapsto \alpha (v):V^{*}\times V\to \F \) is a bilinear pairing.
-
(4) Any real inner product is a bilinear form (what goes wrong for complex inner products?).
-
(5) Let \(A\in M_{m\times n}(\F )\) and define a bilinear pairing \(B_A:\F ^m\times \F ^n\to \F \) by
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B_A(x,y)=\bx ^TA\by .
\end{equation*}
This gives us a new use for matrices.
-
Exercise. Show that \(\bil (U,V;W)\leq W^{U\times V}\). Otherwise said, \(\bil (U,V;W)\) is a vector space under pointwise addition and scalar multiplication.
For the rest of the chapter we focus on the simplest case: bilinear forms \(B:V\times V\to \F \).
5.1.2 Bilinear forms and matrices
-
Definition. Let \(V\) be a vector space over \(\F \) with basis \(\cB =\lst {v}1n\) and let \(B:V\times V\to \F \) be a bilinear form. The matrix of \(B\) with respect to \(\cB \) is \(A\in M_{n\times n}(\F
)\) given by
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
A_{ij}=B(v_i,v_j),
\end{equation*}
for \(\bw 1{i,j}n\).
The matrix \(A\) along with \(\cB \) tells the whole story:
-
Proposition 5.1. Let \(B:V\times V\to \F \) be a bilinear form with matrix \(A\) with respect to \(\cB =\lst {v}1n\). Then
\(B\) is completely determined by \(A\): if \(v=\sum _{i=1}^nx_iv_i\) and \(w=\sum _{j=1}^ny_jv_j\) then
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B(v,w)=\sum _{i,j=1}^nx_iy_jA_{ij},
\end{equation*}
or, equivalently, for all \(x,y\in \F ^n\),
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B(\phi _{\cB }(x),\phi _{\cB }(y))=B_A(x,y)=\bx ^TA\by .
\end{equation*}
-
Proof. We simply expand out using the bilinearity of \(B\):
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B(v,w)=\sum _{i,j=1}^nx_iy_jB(v_i,v_j)=\sum _{i,j=1}^nx_iy_jA_{ij}.
\end{equation*}
□
-
.
-
-
(1) When \(V=\F ^n\) and \(\cB \) is the standard basis (so that \(\phi _{\cB }=\id _{\F ^n}\)), this tells us that any bilinear form on \(V\) is \(B_{A}\) for some matrix \(A\in M_{n\times n}(\F )\).
-
(2) There is a similar analysis for any bilinear map \(B:U\times V\to W\). In this case, \(B\) is determined by \(B(u_i,v_j)\in W\) for \(\lst {u}1m\) a basis of \(U\) and \(\lst {v}1n\) a basis of \(V\).
How does \(A\) change when we change basis of \(V\)?
-
Proposition 5.2. Let \(B:V\times V\to \F \) be a bilinear form with matrices \(A\) and \(A'\) with respect to bases \(\cB
\) and \(\cB '\) of \(V\). Then
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
A'=P^TAP
\end{equation*}
where \(P\) is the change of basis matrix1from \(\cB \) to \(\cB '\): thus \(\phi _P=\phi _{\cB }^{-1}\circ \phi _{\cB '}\).
-
Proof. Since \(\phi _{\cB '}=\phi _{\cB }\circ \phi _{P}\), we have
\(\seteqnumber{0}{5.}{0}\)
\begin{multline*}
\bx ^TA'\by =B(\phi _{\cB '}(x),\phi _{\cB '}(y))=B(\phi _{\cB }(\phi _P(x)),\phi _{\cB }(\phi _P(y)))\\ =B_A(\phi _P(x),\phi _P(y))=(P\bx )^TA(P\by )=\bx ^T(P^TAP)\by ,
\end{multline*}
for all \(x,y\in \F ^n\). Taking \(x=e_i\) and \(y=e_j\), this gives \(A'_{ij}=(P^TAP)_{ij}\) so that \(A'=P^TAP\). □
This prompts:
-
Definition. We say that matrices \(A,B\in M_{n\times n}(\F )\) are congruent if there is \(P\in \GL (n,\F )\) such that
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*}
B=P^TAP.
\end{equation*}