Gramian matrix

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In systems theory and linear algebra, a Gramian matrix is a real-valued 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:

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

The functions are linearly independent if and only if $G$ is nonsingular. Its determinant is known as the Gram determinant or Gramian. It is named for Jørgen Pedersen Gram.

In fact this is a special case of a quantitative measure of linear independence of vectors, available in any Hilbert space. According to that definition, for E a real prehilbert space, if

x₁,..., x_n

are n vectors of E, the associated Gram matrix is the symmetric matrix

(x_i|x_j).

The Gram determinant is the determinant of this matrix,

$G(x_1,\dots, x_n)=\begin{vmatrix} (x_1|x_1) & (x_1|x_2) &\dots & (x_1|x_n)\\ (x_2|x_1) & (x_2|x_2) &\dots & (x_2|x_n)\\ \vdots&\vdots&&\vdots\\ (x_n|x_1) & (x_n|x_2) &\dots & (x_n|x_n)\end{vmatrix}$