一般化ヘロンの公式：Cayley-Menger行列式

はじめに

　この記事ではヘロンの公式を$n$次元に一般化した公式を紹介します。
　まずヘロンの公式とは以下の公式のことを言うのでした。

　そしてこれを$n$次元に一般化した公式：ケイリー・メンガー行列式は以下のようになります。

　実際に例えば$n=2$(三角形)のときにこれを計算すると
$$\begin{array}{rcl}{S}^2& =& -\frac{1}{2^2\cdot 2^2}\det \left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& a^2& b^2\\ 1& a^2& 0& c^2\\ 1& b^2& c^2& 0\end{array}\right)\\ & =& \frac{1}{2^4}((a^2{b}^2+b^2{c}^2+c^2{a}^2)-(a^4+b^4+c^4))\\ & =& s(s-a)(s-b)(s-c)\end{array}$$
とヘロンの公式が得られたり、
$$S^2=\frac{1}{2^2}\det \left(\begin{array}{cc}\alpha \cdot \alpha & \alpha \cdot \beta \\ \beta \cdot \alpha & \beta \cdot \beta \end{array}\right)=\frac{1}{2^2}(|\alpha {|}^2|\beta {|}^2-(\alpha \cdot \beta {)}^2)$$
とよく知られた公式も出てきます。
　ちなみに$n=3$(三角錐)のときは
$$\begin{array}{rcl}36V^2& =& \det \left(\begin{array}{ccc}|{\mathit{x}}_1{|}^2& {\mathit{x}}_1\cdot {\mathit{x}}_2& {\mathit{x}}_1\cdot {\mathit{x}}_3\\ {\mathit{x}}_1\cdot {\mathit{x}}_2& |{\mathit{x}}_2{|}^2& {\mathit{x}}_2\cdot {\mathit{x}}_3\\ {\mathit{x}}_1\cdot {\mathit{x}}_3& {\mathit{x}}_2\cdot {\mathit{x}}_3& |{\mathit{x}}_3{|}^2\end{array}\right)\\ & =& |{\mathit{x}}_1{|}^2|{\mathit{x}}_2{|}^2|{\mathit{x}}_3{|}^2-|{\mathit{x}}_1{|}^2({\mathit{x}}_2\cdot {\mathit{x}}_3{)}^2-|{\mathit{x}}_2{|}^2({\mathit{x}}_3\cdot {\mathit{x}}_1{)}^2-|{\mathit{x}}_3{|}^3({\mathit{x}}_1\cdot {\mathit{x}}_2{)}^2+2({\mathit{x}}_1\cdot {\mathit{x}}_2)({\mathit{x}}_2\cdot {\mathit{x}}_3)({\mathit{x}}_3\cdot {\mathit{x}}_1)\end{array}$$
なんていう公式が成り立ったりします。

証明

　あとは
$$V^{\prime }=|\det \left(\begin{array}{cccc}{\mathit{x}}_1& {\mathit{x}}_2& \cdots & {\mathit{x}}_n\end{array}\right)|$$
を変形していく。
$$\begin{array}{rcl}{V}^2& =& {\left(\frac{V^{\prime }}{n!}\right)}^2\\ & =& \frac{1}{(n!{)}^2}\det \left({\left(\begin{array}{cccc}{\mathit{x}}_1& {\mathit{x}}_2& \cdots & {\mathit{x}}_n\end{array}\right)}^t\left(\begin{array}{cccc}{\mathit{x}}_1& {\mathit{x}}_2& \cdots & {\mathit{x}}_n\end{array}\right)\right)\\ & =& \frac{1}{(n!{)}^2}\det {({\mathit{x}}_i\cdot {\mathit{x}}_j)}_{1\le i,j\le n}\cdots (\ast )\\ & =& \frac{1}{2^n(n!{)}^2}\det (2{\mathit{x}}_i\cdot {\mathit{x}}_j)\\ & =& \frac{1}{2^n(n!{)}^2}\det (d_{0,i}^2+d_{0,j}^2-d_{i,j}^2)\\ & =& \frac{(-1{)}^n}{2^n(n!{)}^2}\det (d_{i,j}^2-d_{0,i}^2-d_{0,j}^2)\\ & =& \frac{(-1{)}^{3n+3}}{2^n(n!{)}^2}\det \left(\begin{array}{cccccc}0& 0& 0& \cdots & 0& 1\\ 0& -2d_{0,1}^2& d_{1,2}^2-d_{0,1}^2-d_{0,2}^2& \cdots & d_{1,n}^2-d_{0,1}^2-d_{0,n}^2& 0\\ 0& d_{1,2}^2-d_{0,1}^2-d_{0,2}^2& -2d_{0,2}^2& \cdots & d_{2,n}^2-d_{0,2}^2-d_{0,n}^2& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& d_{1,n}^2-d_{0,1}^2-d_{0,n}^2& d_{2,n}^2-d_{0,2}^2-d_{0,n}^2& \cdots & -2d_{0,n}^2& 0\\ 1& 0& 0& \cdots & 0& 0\end{array}\right)\\ & =& \frac{(-1{)}^{n+1}}{2^n(n!{)}^2}\det \left(\begin{array}{cccccc}0& d_{0,1}^2& d_{0,2}^2& \cdots & d_{0.n}^2& 1\\ d_{0,1}^2& -2d_{0,1}^2& d_{1,2}^2-d_{0,1}^2-d_{0,2}^2& \cdots & d_{1,n}^2-d_{0,1}^2-d_{0,n}^2& 0\\ d_{0,2}^2& d_{1,2}^2-d_{0,1}^2-d_{0,2}^2& -2d_{0,2}^2& \cdots & d_{2,n}^2-d_{0,2}^2-d_{0,n}^2& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ d_{0,n}^2& d_{1,n}^2-d_{0,1}^2-d_{0,n}^2& d_{2,n}^2-d_{0,2}^2-d_{0,n}^2& \cdots & -2d_{0,n}^2& 0\\ 1& 0& 0& \cdots & 0& 0\end{array}\right)\\ & =& \frac{(-1{)}^{n+1}}{2^n(n!{)}^2}\det \left(\begin{array}{cccccc}0& d_{0,1}^2& d_{0,2}^2& \cdots & d_{0.n}^2& 1\\ d_{0,1}^2& 0& d_{1,2}^2& \cdots & d_{1,n}^2& 1\\ d_{0,2}^2& d_{1,2}^2& 0& \cdots & d_{2,n}^2& 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ d_{0,n}^2& d_{1,n}^2& d_{2,n}^2& \cdots & 0& 1\\ 1& 1& 1& \cdots & 1& 0\end{array}\right)\\ & =& \frac{(-1{)}^{n+1}}{2^n(n!{)}^2}\det \left(\begin{array}{cccccc}0& 1& 1& 1& \cdots & 1\\ 1& 0& d_{0,1}^2& d_{0,2}^2& \cdots & d_{0,n}^2\\ 1& d_{0,1}^2& 0& d_{1,2}^2& \cdots & d_{1,n}^2\\ 1& d_{0,2}^2& d_{1,2}^2& 0& \cdots & d_{2,n}^2\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& d_{0,n}^2& d_{1,n}^2& d_{2,n}^2& \cdots & 0\end{array}\right)\end{array}$$
といった具合である。