Editing Gram matrix (section)

==Gram determinant==
The '''Gram determinant''' or '''Gramian''' is the determinant of the Gram matrix:
<math display=block>\bigl|G(v_1, \dots, v_n)\bigr| = \begin{vmatrix}
  \langle v_1,v_1\rangle & \langle v_1,v_2\rangle &\dots & \langle v_1,v_n\rangle \\
  \langle v_2,v_1\rangle & \langle v_2,v_2\rangle &\dots & \langle v_2,v_n\rangle \\
  \vdots & \vdots & \ddots & \vdots \\
  \langle v_n,v_1\rangle & \langle v_n,v_2\rangle &\dots & \langle v_n,v_n\rangle
\end{vmatrix}.</math>

If <math>v_1, \dots, v_n</math> are vectors in <math>\mathbb{R}^m</math> then it is the square of the ''n''-dimensional volume of the [[Parallelepiped#Parallelotope|parallelotope]] formed by the vectors. In particular, the vectors are [[Linear independence|linearly independent]] [[if and only if]] the parallelotope has nonzero ''n''-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is [[Non-singular matrix|nonsingular]]. When {{nowrap|''n'' > ''m''}} the determinant and volume are zero.  When {{nowrap|1=''n'' = ''m''}}, this reduces to the standard theorem that the absolute value of the determinant of ''n'' ''n''-dimensional vectors is the ''n''-dimensional volume. The volume of the [[simplex]] formed by the vectors is {{math|Volume(parallelotope) / ''n''!}}.

When <math>v_1, \dots, v_n</math> are linearly independent, the distance between a point <math>x</math> and the linear span of <math>v_1, \dots, v_n</math> is <math>\sqrt{\frac{|G(x,v_1, \dots, v_n)|}{|G(v_1, \dots, v_n)|}}</math>.

Consider the moment problem: given <math>c_1, \dots, c_n \in \mathbb C</math>, find a vector <math>v</math> such that <math display="inline">\left\langle v, v_i\right\rangle=c_i</math>, for all <math display="inline">1 \leqslant i \leqslant n</math>. There exists a unique solution with minimal norm:<ref>{{Cite journal |last1=Ramon |first1=Garcia, Stephan |last2=Javad |first2=Mashreghi |last3=T. |first3=Ross, William |date=2023-01-30 |title=Operator Theory by Example |url=https://academic.oup.com/book/45766 |journal=OUP Academic |language=en |doi=10.1093/o|doi-broken-date=13 April 2025 }}</ref>{{Pg|page=38}}<math display="block">v=-\frac{1}{G\left(v_1, v_2, \ldots, v_n\right)} \det
\begin{bmatrix}
0 & c_1 & c_2 & \cdots & c_n \\
v_1 & \left\langle v_1, v_1\right\rangle & \left\langle v_1, v_2\right\rangle & \cdots & \left\langle v_1, v_n\right\rangle \\
v_2 & \left\langle v_2, v_1\right\rangle & \left\langle v_2, v_2\right\rangle & \cdots & \left\langle v_2, v_n\right\rangle \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
v_n & \left\langle v_n, v_1\right\rangle & \left\langle v_n, v_2\right\rangle & \cdots & \left\langle v_n, v_n\right\rangle
\end{bmatrix}</math>The Gram determinant can also be expressed in terms of the [[exterior product]] of vectors by
:<math>\bigl|G(v_1, \dots, v_n)\bigr| = \| v_1 \wedge \cdots \wedge v_n\|^2.</math>

The Gram determinant therefore supplies an [[exterior product#inner product|inner product]] for the space {{tmath|{\textstyle\bigwedge}^{\!n}(V)}}. If an [[orthonormal basis]] ''e''<sub>''i''</sub>, {{nowrap|1=''i'' = 1, 2, ..., ''n''}} on {{tmath|V}} is given, the vectors 
: <math> e_{i_1} \wedge \cdots \wedge e_{i_n},\quad i_1 < \cdots < i_n, </math>
will constitute an orthonormal basis of ''n''-dimensional volumes on the space {{tmath|{\textstyle\bigwedge}^{\!n}(V)}}. Then the Gram determinant <math>\bigl|G(v_1, \dots, v_n)\bigr|</math> amounts to an ''n''-dimensional [[Pythagorean Theorem#Sets_of_m-dimensional_objects_in_n-dimensional_space|Pythagorean Theorem]] for the volume of the parallelotope formed by the vectors <math>v_1 \wedge \cdots \wedge v_n</math> in terms of its projections onto the basis volumes <math>e_{i_1} \wedge \cdots \wedge e_{i_n}</math>.

When the vectors <math>v_1, \ldots, v_n \in \mathbb{R}^m</math> are defined from the positions of points <math>p_1, \ldots, p_n</math> relative to some reference point <math>p_{n+1}</math>,
:<math display="block">(v_1, v_2, \ldots, v_n) = (p_1 - p_{n+1}, p_2 - p_{n+1}, \ldots, p_n - p_{n+1})\,,</math>
then the Gram determinant can be written as the difference of two Gram determinants,
:<math display=block>
\bigl|G(v_1, \dots, v_n)\bigr| = \bigl|G((p_1, 1), \dots, (p_{n+1}, 1))\bigr| - \bigl|G(p_1, \dots, p_{n+1})\bigr|\,,
</math>
where each <math>(p_j, 1)</math> is the corresponding point <math>p_j</math> supplemented with the coordinate value of 1 for an <math>(m+1)</math>-st dimension.{{Citation needed|reason=This relation between Gram matrices is apparently true but needs a citation to support its [[WP:N|notability]].|date=February 2022}} Note that in the common case that {{math|1=''n'' = ''m''}}, the second term on the right-hand side will be zero.