matrix representation of a bilinear form

Given a bilinear formPlanetmathPlanetmath, B:U×VK, 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, 1={e1,} and 2={f1,}. Now we define the matrix C with entries Cij=B(ei,fj). This will be the matrix associated to B with respect to this basis as follows; If we write x,yV as column vectorsMathworldPlanetmath in terms of the chosen bases, then check B(x,y)=xTCy. Further if we choose the corresponding dual bases for U and V then C and CT are the corresponding matrices for BR and BL, respectively (in the sense of linear maps). Thus we see that a symmetric bilinear formMathworldPlanetmath is represented by a symmetric matrixMathworldPlanetmath, and similarly for skew-symmetric forms.

Let 1 and 2 be new bases, and P and Q the corresponding change of basis matrices. Then the new matrix is C=PTCQ.

Title matrix representationPlanetmathPlanetmath 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