Let \(V\) be a finite-dimensional vector space over \(\R\text{.}\) (We will use a real vector space for simplicity, but most of this works just as well for complex vector spaces.) Recall from SectionΒ 3.5 that a linear functional on \(V\) is a linear map \(\phi:V\to \R\text{.}\)
Caution: the bilinear form \(\phi\) is not assumed to be symmetric, so the linear functionals \(\eta\) and \(\psi\) defined above are (in general) different functions.
An example of a bilinear form is the dot product on \(\R^n\text{.}\) Given \(\vv,\ww\in\R^n\text{,}\) the function \(\phi(\vv,\ww) = \vv\dotp\ww\) defines a bilinear form.
Let \(V\) be a finite-dimensional vector space, and let \(B=\basis{e}{n}\) be an ordered basis for \(V\text{.}\) Let \(\phi:V\times V\to \R\) be a bilinear form on \(V\text{.}\)
The above exercise tells us that we can study bilinear forms on a vector space by studying their matrix representations. This depends on a choice of basis, but, as one might expect, matrix representations with respect to different bases are similar.
Let \(B_1,B_2\) be two ordered bases for a finite-dimensional vector space \(V\text{,}\) and let \(P=P_{B_1\leftarrow B_2}\) be the change of basis matrix for these bases. Let \(\phi:V\times V\to \R\) be a bilinear form on \(V\text{.}\)
If \(A_\phi\) is the matrix of \(\phi\) with respect to the basis \(B_1\text{,}\) show that the matrix of \(\phi\) with respect to \(B_2\) is equal to \(P^TA_\phi P\text{.}\)
A bilinear form \(\phi\) on \(V\) is symmetric if \(\phi(\vv,\ww)=\phi(\ww,\vv)\) for all \(\vv,\ww\in V\text{,}\) and antisymmetric (or alternating) if \(\phi(\vv,\ww)=-\phi(\ww,\vv)\) for all \(\vv,\ww\) in \(V\text{.}\)
A bilinear form is nondegenerate if, for each nonzero vector \(\vv\in V\text{,}\) there exists a vector \(\ww\in V\) such that \(\phi(\vv,\ww)\neq 0\text{.}\) (Alternatively, for each nonzero \(\vv\in V\text{,}\) the linear functional \(\alpha(\ww)=\phi(\vv,\ww)\) is nonzero.)
Two types of bilinear forms are of particular importance: a symmetric, nondegenerate bilinear form on \(V\) is called an inner product on \(V\text{,}\) if it is also positive-definite: for each \(\vv\in V\text{,}\)\(\phi(\vv,\vv)\geq 0\text{,}\) with equality only if \(\vv=\mathbf{0}\text{.}\) Inner products are a generalization of the dot product from ChapterΒ 3. A future version of this book may take the time to study inner products in more detail, but for now we will look at another type of bilinear form.
A nondegenerate, antisymmetric bilinear form \(\omega\) on \(V\) is called a linear symplectic structure on \(V\text{,}\) and we call the pair \((V,\omega)\) a symplectic vector space. Symplectic structures are important in differential geometry and mathematical physics. (They can be used to encode Hamiltonβs equations in classical mechanics.)
A more general example is given by \(V=\R^{2n}\text{.}\) If we write \(\vv = \langle a_1,b_1,\ldots, a_n,b_n\rangle\text{,}\)\(\ww = \langle c_1,d_1,\ldots, c_n,d_n\rangle\text{,}\) then the standard symplectic structure on \(\R^{2n}\) is given by
A theorem that you will not be asked to prove (itβs a long proof...) is that if a vector space \(V\) has a linear symplectic structure \(\omega\text{,}\) then the dimension of \(V\) is even, and \(V\) has a basis \(\{\mathbf{e}_1,\mathbf{f}_1,\ldots, \mathbf{e}_n,\mathbf{f}_n\}\) with respect to which the matrix representation of \(\omega\) is equivalent to the standard symplectic structure on \(\R^{2n}\text{.}\)
For the symplectic structure \(\omega(\vv,\ww) = v_1w_2-v_2w_1\) on \(\R^2\text{,}\) as given above, show that the matrix of \(\omega\) with respect to the standard basis is the matrix \(J_1 = \bbm 0\amp -1\\ 1\amp 0\ebm\text{.}\)
Then, for any symplectic vector space \((V,\omega)\text{,}\) show that, with respect to the basis \(\{\mathbf{e}_1,\mathbf{f}_1,\ldots, \mathbf{e}_n,\mathbf{f}_n\}\) described in RemarkΒ 5.3.5 above, the matrix of \(\omega\) has the block form
Let \(\langle \vv,\ww\rangle\) denote the standard complex inner product on \(\C^n\text{.}\) (Recall that such an inner product is complex linear in the second argument, but for any complex scalar \(c\text{,}\)\(\langle c\vv,\ww\rangle = \overline{c}\langle \vv,\ww\rangle\text{.}\))
For more reading on multilinear forms and determinants, see the 4th edition of Linear Algebra Done Right, by Sheldon Axler. For more reading on linear symplectic structures, see First Steps in Differential Geometry, by Andrew McInerney.