Gram determinant

Gram determinant

[′gram di′tərm·ə·nənt]
(mathematics)
The Gram determinant of vectorsv1, …,vn from an inner product space is the determinant of the n × n matrix with the inner product ofvi andvj as entry in the i th column and j th row; its vanishing is a necessary and sufficient condition for linear dependence.
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The Gram determinant [2] (composed from pairwise scalar products of the current and voltage IV vectors) equals to square of the vectorial IP (8)
The Gram determinant at each moment of time quadratically complements the scalar IP to the total (apparent) IP (6)
(2) Given [x.sub.1], [x.sub.2], ..., [x.sub.k] the (k + 1)st point [x.sub.k+1] [member of] K is chosen so that the Gram determinant,
and hence the Gram determinant is maximized at x = 0 and [x.sub.3] = 0.
Then the absolute value of the Gram determinant of {[[eta].sub.i1],...,[[eta].sub.in]} is independent of i1,...,in, where {i1,...,in} is a subset of {1,...,n + 1}
A referee suggested us that Theorem 1.1 can be proven by another method, i.e., using Wilson's result [16] on the Gram determinants for the Askey-Wilson polynomials.
Orthogonal functions from gram determinants. SIAMJ.