# matrix representation of a bilinear form

Given a bilinear form^{}, $B:U\times V\to K$, 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, ${\mathcal{B}}_{1}=\{{e}_{1},\mathrm{\dots}\}$ and ${\mathcal{B}}_{2}=\{{f}_{1},\mathrm{\dots}\}$. 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 ${\mathcal{B}}_{1}^{\prime}$ and ${\mathcal{B}}_{2}^{\prime}$ 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 |