Cayley-Menger Determinant

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Definition

A determinant that gives the volume of a simplex in $j$ dimensions.


Theorem

Let $S$ be a $j$-simplex in $\R^n$.

Let the vertices of $S$ be $v_1, v_2, \ldots, v_{j+1}$.

Let $B = \left[{\beta_{ij} }\right]$ denote the $\left({j + 1}\right) \times \left({j + 1}\right)$ matrix given by:

$\beta_{ij} = \left\vert{v_i - v_j}\right\vert_2^2$



Then the content $V_j$ is given by:

\(\displaystyle V_j^2 \left({S}\right)\) \(=\) \(\displaystyle \frac {\left({-1}\right)^{j+1} } {2^j \left({j!}\right)^2} \det C\) $\quad$ $\quad$

where $C$ is the $\left({j + 2}\right) \times \left({j + 2}\right)$ matrix obtained from $B$ by bordering $B$ with a top row $\left({0, 1, \ldots, 1}\right)$ and a left column $\left({0, 1, \ldots, 1}\right)^\intercal$.


Proof


Also see


Source of Name

This entry was named for Arthur Cayley and Karl Menger.


Sources

Colins, Karen D.. "Cayley-Menger Determinant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cayley-MengerDeterminant.html