Gram determinant

Let V be an inner product spaceMathworldPlanetmath over a field k with , the inner productMathworldPlanetmath on V (note: since k is not restricted to be either or , the inner product here shall mean a symmetric bilinear formMathworldPlanetmath on V). Let x1,x2,,xn be arbitrary vectors in V. Set rij=xi,xj. The Gram determinantMathworldPlanetmath of x1,x2,,xn is defined to be the determinantMathworldPlanetmath of the symmetric matrixMathworldPlanetmath


Let’s denote this determinant by Gram[x1,x2,,xn].


  1. 1.

    Gram[x1,,xi,,xj,,xn]=Gram[x1,,xj,,xi,,xn]. More generally, Gram[x1,,xn]=Gram[xσ(1),,xσ(n)], where σ is a permutationMathworldPlanetmath on {1,,n}.

  2. 2.

    Gram[x1,,axi+bxj,,xj,,xn]=a2Gram[x1,,xi,,xj,,xn], a,bk.

  3. 3.

    Setting a=0 and b=1 in Property 2, we get Gram[x1,,xj,,xj,,xn]=0.

  4. 4.

    Properties 2 and 3 can be generalized as follows: if xi (in the ith term) is replaced by a linear combinationMathworldPlanetmath y=r1x1++rnxn, then

  5. 5.

    Suppose k is an ordered field. Then it can be shown that the Gram determinant is at least 0, and at most the product x1,x1xn,xn.

  6. 6.

    Suppose that in addition to k being ordered, that every positive elementMathworldPlanetmathPlanetmathPlanetmath in k is a square, then the Gram determinant is equal to the square of the volume of the (hyper)parallelepipedMathworldPlanetmath generated by x1,,xn. (Recall that an n-dimensional parallelepiped is the set of vectors which are linear combinations of the form r1x1++rnxn where 0ri1.)

  7. 7.

    It’s now easy to see that in Property 5, the Gram determinant is 0 if the xi’s are linearly dependent, and attains its maximum if the xi’s are pairwise orthogonalMathworldPlanetmathPlanetmathPlanetmath (a quick proof: in the above matrix, rij=0 if ij), which corresponds exactly to the square of the volume of the hyperparallelepiped spanned by the xi’s.

  8. 8.

    If e1,,en are basis elements of a quadratic space V over an order field whose positive elements are squares, then V is , or , iff Gram[e1,,en]=0.


  • 1 Georgi E. Shilov, “An Introduction to the Theory of Linear SpacesPlanetmathPlanetmath”, translated from Russian by Richard A. Silverman, 2nd Printing, Prentice-Hall, 1963.
Title Gram determinant
Canonical name GramDeterminant
Date of creation 2013-03-22 15:41:37
Last modified on 2013-03-22 15:41:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Related topic GrammianDeterminant
Related topic GramMatrix