matrix representation of a bilinear form

Given a bilinear form, $B:U\times V\rightarrow K$ , we show how we can represent it with a matrix, with respect to a particular pair of bases for $U$ and $V$

Suppose $U$ and $V$ are finite-dimensional and we have chosen bases, ${{\cal B}_{1}}=\{e_{1},\ldots\}$ and ${{\cal B}_{2}}=\{f_{1},\ldots\}$ . Now we define the matrix $C$ with entries $C_{ij}=B(e_{i},f_{j})$ . This will be the matrix associated to $B$ with respect to this basis as follows; If we write $x,y\in V$ as column vectors in terms of the chosen bases, then check $B(x,y)=x^{T}Cy$ . Further if we choose the corresponding dual bases for $U^{\ast}$ and $V^{\ast}$ then $C$ and $C^{T}$ are the corresponding matrices for $B_{R}$ and $B_{L}$ , respectively (in the sense of linear maps). Thus we see that a symmetric bilinear form is represented by a symmetric matrix, and similarly for skew-symmetric forms.

Let ${{\cal B}_{1}^{\prime}}$ and ${{\cal B}_{2}^{\prime}}$ be new bases, and $P$ and $Q$ the corresponding change of basis matrices. Then the new matrix is $C^{\prime}=P^{T}CQ$ .

Title	matrix representation of a bilinear form
Canonical name	MatrixRepresentationOfABilinearForm
Date of creation	2013-03-22 14:56:22
Last modified on	2013-03-22 14:56:22
Owner	vitriol (148)
Last modified by	vitriol (148)
Numerical id	5
Author	vitriol (148)
Entry type	Definition
Classification	msc 15A63
Classification	msc 11E39
Classification	msc 47A07