Gramian matrix
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In systems theory and linear algebra, the Gramian matrix of a set of functions 
is a real-valued symmetric matrix G = [Gij], where 
.
The Gramian matrix can be used to test for linear independence of functions. Namely, 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
are n vectors of E, the associated Gram matrix is the symmetric matrix
.
The Gram determinant is the determinant of this matrix,
All eigenvalues of a Gramian matrix are real and non-negative and the matrix is thus also positive semidefinite.
