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:
- $\beta_{ij} = \left\vert{v_i - v_j}\right\vert_2^2$
where $\left\vert{v_i - v_j}\right\vert_2$ is the vector 2-norm of the vector $v_i - v_j$.
Then the content $V_j$ is given by:
| \(\ds V_j^2 \left({S}\right)\) | \(=\) | \(\ds \frac {\left({-1}\right)^{j+1} } {2^j \left({j!}\right)^2} \det C\) |
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
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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