Gramian matrix

In systems theory and linear algebra, a Gramian matrix is a real symmetric matrix that can be used to test for linear independence of functions. The Gramian matrix of a set of functions $\{l_i(\cdot),\,i=1,\dots,n\}$ is defined as

$G=[G_{ij}],\,\,G_{ij}=\int_{t_0}^{t_f} l_i(\tau)l_j(\tau)\, d\tau$

If the functions are linearly independent, then $G$ is nonsingular. Its determinant is known as the Gram determinant.

In fact this is a special case of a quantitative measure of linear independence of vectors, available in any Hilbert space.

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07-14-2008 23:18:10
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