MA20216: Algebra 2A

Chapter 6 Bilinearity

We give an introduction to a general theory of “multiplication” of vectors.

6.1 Bilinear maps

6.1.1 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:
\(\seteqnumber{0}{6.}{0}\)
\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 \(V\).

Remark. 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,
\(\seteqnumber{0}{6.}{0}\)
\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:
  \(\seteqnumber{0}{6.}{0}\)
  \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:
  \(\seteqnumber{0}{6.}{0}\)
  \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
  \(\seteqnumber{0}{6.}{0}\)
  \begin{equation*} B_A(x,y)=\bx ^TA\by . \end{equation*}
  
  This gives us a new use for matrices.

Notation. We let \(\bil (U,V;W)\) denote the set of bilinear maps \(U\times V\to W\).

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 \(B:V\times V\to \F \).

6.1.2 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
\(\seteqnumber{0}{6.}{0}\)
\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 6.1. 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
\(\seteqnumber{0}{6.}{0}\)
\begin{equation*} B(v,w)=\sum _{i,j=1}^nx_iy_jA_{ij}, \end{equation*}

or, equivalently, for all \(x,y\in \F ^n\),
\(\seteqnumber{0}{6.}{0}\)
\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 \(B\):
\(\seteqnumber{0}{6.}{0}\)
\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*}

□

Remarks.
- (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 6.2. 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
\(\seteqnumber{0}{6.}{0}\)
\begin{equation*} A'=P^TAP \end{equation*}

where \(P\) is the change of basis matrix¹from \(\cB \) to \(\cB '\): thus \(\phi _P=\phi _{\cB }^{-1}\circ \phi _{\cB '}\).

¹ Algebra 1B, Definition 1.6.1.

Proof. Since \(\phi _{\cB '}=\phi _{\cB }\circ \phi _{P}\), we have
\(\seteqnumber{0}{6.}{0}\)
\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
\(\seteqnumber{0}{6.}{0}\)
\begin{equation*} B=P^TAP. \end{equation*}