The Gram matrix 
of real vectors 
, ..., 
in 
is the matrix of their inner
products,
 |
(1)
|
where the superscript 
denotes the transpose. If 
has these vectors as its columns,
then
 |
(2)
|
Consequently, 
is a positive semidefinite matrix,
its matrix rank is the dimension
of the span of the vectors,
and it is a positive definite matrix iff the vectors are linearly
independent. The Gram matrix determines the vectors
in 
up to an orthogonal
transformation.
Gram matrices also give a criterion for realizing prescribed distances in Euclidean space. Let 
for 
, ..., 
, with 
, and form
 |
(3)
|
where 
,
..., 
. The prescribed distances
can be realized by points 
, ..., 
in a Euclidean space iff 
is a positive semidefinite matrix.
Its matrix rank is the smallest possible dimension
of the affine hull of a realization (Young and Householder
1938). For a realization, this is the Gram matrix of 
. Conversely, factoring 
with a real matrix 
provides coordinates by taking 
and the columns of 
as the other points.
For example, prescribe distances between five labels so that 
and every other distance between distinct labels is
1. Every triple is realizable as either an equilateral
triangle of side 1 or a triangle
with sides of lengths 1, 1, 
. Both have area 
. Nevertheless, the corresponding matrix
is
![G=1/2[2 -1 1 1; -1 2 1 1; 1 1 2 1; 1 1 1 2],](/p/https/mathworld.wolfram.com/images/equations/GramMatrix/NumberedEquation4.svg) |
(4)
|
and its determinant is 
. It is therefore not a positive
semidefinite matrix, so no such five points exist in
any Euclidean space. Trott (2006) considers systems
prescribing equal triangle areas of a four-dimensional
simplex. The example shows why solutions of such algebraic
systems must also be tested for Euclidean realizability.
