Grammian determinant

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

\langle f_{i}|f_{j}\rangle=\int_{I}f_{i}f_{j}

It is defined as:

G(f_{1},f_{2},\ldots,f_{n})=\left\lvert\begin{array}[]{@{}cccc@{}}\int_{I}(f_{% 1})^{2}&\int_{I}f_{1}f_{2}&\cdots&\int_{I}f_{1}f_{n}\\ \int_{I}f_{2}f_{1}&\int_{I}(f_{2})^{2}&\cdots&\int_{I}f_{2}f_{n}\\ \vdots&\vdots&\ddots&\vdots\\ \int_{I}f_{n}f_{1}&\int_{I}f_{n}f_{2}&\cdots&\int_{I}(f_{n})^{2}\\ \end{array}\right\rvert

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