# Grammian determinant

The Grammian determinant provides a necessary and sufficient method of determining whether a set of continuous functions^{} ${f}_{1},{f}_{2},\mathrm{\dots},{f}_{n}$ is linearly independent^{} on an interval $I=[a,b]$ with respect to the inner product^{}

$$\u27e8{f}_{i}|{f}_{j}\u27e9={\int}_{I}{f}_{i}{f}_{j}$$ |

It is defined as:

$$G({f}_{1},{f}_{2},\mathrm{\dots},{f}_{n})=\left|\begin{array}{cccc}\hfill {\int}_{I}{({f}_{1})}^{2}\hfill & \hfill {\int}_{I}{f}_{1}{f}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\int}_{I}{f}_{1}{f}_{n}\hfill \\ \hfill {\int}_{I}{f}_{2}{f}_{1}\hfill & \hfill {\int}_{I}{({f}_{2})}^{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\int}_{I}{f}_{2}{f}_{n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\int}_{I}{f}_{n}{f}_{1}\hfill & \hfill {\int}_{I}{f}_{n}{f}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\int}_{I}{({f}_{n})}^{2}\hfill \end{array}\right|$$ |

If the functions are continuous on $I$, then $G=0$ if and only if the set of functions is linearly dependent. Note that the Grammian determinant is a special case of the more general Gram determinant^{}.

Title | Grammian determinant |
---|---|

Canonical name | GrammianDeterminant |

Date of creation | 2013-03-22 17:37:33 |

Last modified on | 2013-03-22 17:37:33 |

Owner | slider142 (78) |

Last modified by | slider142 (78) |

Numerical id | 6 |

Author | slider142 (78) |

Entry type | Definition |

Classification | msc 34A12 |

Related topic | WronskianDeterminant |

Related topic | GramDeterminant |