Let be an inner product space over a field with the inner product on (note: since is not restricted to be either or , the inner product here shall mean a symmetric bilinear form on ). Let be arbitrary vectors in . Set . The Gram determinant of is defined to be the determinant of the symmetric matrix
Let’s denote this determinant by .
. More generally, , where is a permutation on .
Setting and in Property 2, we get .
Properties 2 and 3 can be generalized as follows: if (in the th term) is replaced by a linear combination , then
Suppose is an ordered field. Then it can be shown that the Gram determinant is at least 0, and at most the product .
Suppose that in addition to being ordered, that every positive element in is a square, then the Gram determinant is equal to the square of the volume of the (hyper)parallelepiped generated by . (Recall that an -dimensional parallelepiped is the set of vectors which are linear combinations of the form where .)
It’s now easy to see that in Property 5, the Gram determinant is 0 if the ’s are linearly dependent, and attains its maximum if the ’s are pairwise orthogonal (a quick proof: in the above matrix, if ), which corresponds exactly to the square of the volume of the hyperparallelepiped spanned by the ’s.
If are basis elements of a quadratic space over an order field whose positive elements are squares, then is , or , iff .
- 1 Georgi E. Shilov, “An Introduction to the Theory of Linear Spaces”, translated from Russian by Richard A. Silverman, 2nd Printing, Prentice-Hall, 1963.
|Date of creation||2013-03-22 15:41:37|
|Last modified on||2013-03-22 15:41:37|
|Last modified by||CWoo (3771)|