# diagonal quadratic form

Let $Q(\bm{x})\in k[{x}_{1},\mathrm{\dots},{x}_{n}]$ be a quadratic form^{} over a field $k$ ($\mathrm{char}(k)\ne 2$), where $\bm{x}$ is the column vector^{} ${({x}_{1},\mathrm{\dots},{x}_{n})}^{T}$. We write $Q$ as

$$Q(\bm{x})={\bm{x}}^{T}M(Q)\bm{x},$$ |

where $M(Q)$ is the associated $n\times n$ symmetric matrix^{} over $k$. We say that $Q$ is a *diagonal quadratic form ^{}* if $M(Q)$ is a diagonal matrix

^{}.

Let’s see what a diagonal quadratic form looks like. If $M=M(Q)$ is diagonal whose diagonal entry in cell $(i,i)$ is ${r}_{i}$, then

$Q(\bm{x})={\bm{x}}^{T}\left(\begin{array}{ccc}\hfill {r}_{1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {r}_{n}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {x}_{n}\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {x}_{1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{n}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {r}_{1}{x}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {r}_{n}{x}_{n}\hfill \end{array}\right)={r}_{1}{x}_{1}^{2}+\mathrm{\cdots}+{r}_{n}{x}_{n}^{2}.$

So the coefficients of ${x}_{i}{x}_{j}$ for $i\ne j$ are all $0$ in a diagonal quadratic form. A diagonal quadratic form is completely determined by the diagonal entries of $M(Q)$.

Remark. Every quadratic form is equivalent^{} (http://planetmath.org/EquivalentQuadraticForms) to a diagonal quadratic form. On the other hand, a quadratic form may be to more than one diagonal quadratic form.

Title | diagonal quadratic form |
---|---|

Canonical name | DiagonalQuadraticForm |

Date of creation | 2013-03-22 15:42:05 |

Last modified on | 2013-03-22 15:42:05 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 11E81 |

Classification | msc 15A63 |

Classification | msc 11H55 |

Synonym | canonical quadratic form |

Related topic | DiagonalizationOfQuadraticForm |