ビネ・コーシーの恒等式
∑ 1 ≤ i < j ≤ n ( a i b j − a j b i ) ( c i d j − c j d i ) = ∑ 1 ≤ i < j ≤ n ( a i c i b j d j + a j c j b i d i ) + ∑ i = 1 n a i c i b i d i − ∑ 1 ≤ i < j ≤ n ( a i d i b j c j + a j d j b i c i ) − ∑ i = 1 n a i d i b i c i = ( ∑ 1 ≤ i < j ≤ n + ∑ 1 ≤ j < i ≤ n + ∑ 1 ≤ i = j ≤ n ) a i c i b j d j − ( ∑ 1 ≤ i < j ≤ n + ∑ 1 ≤ j < i ≤ n + ∑ 1 ≤ i = j ≤ n ) a i d i b j c j = ∑ 1 ≤ i ≤ n 1 ≤ j ≤ n a i c i b j d j − ∑ 1 ≤ i ≤ n 1 ≤ j ≤ n a i d i b j c j = ( ∑ i = 1 n a i c i ) ( ∑ j = 1 n b j d j ) − ( ∑ i = 1 n a i d i ) ( ∑ j = 1 n b j c j ) {\displaystyle {\begin{aligned}&\textstyle \sum \limits _{1\leq i<j\leq n}(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})\\&=\textstyle \sum \limits _{1\leq i<j\leq n}(a_{i}c_{i}b_{j}d_{j}+a_{j}c_{j}b_{i}d_{i})+\sum \limits _{i=1}^{n}a_{i}c_{i}b_{i}d_{i}-\sum \limits _{1\leq i<j\leq n}(a_{i}d_{i}b_{j}c_{j}+a_{j}d_{j}b_{i}c_{i})-\sum \limits _{i=1}^{n}a_{i}d_{i}b_{i}c_{i}\\&=\left(\textstyle \sum \limits _{1\leq i<j\leq n}+\sum \limits _{1\leq j<i\leq n}+\sum \limits _{1\leq i=j\leq n}\right)a_{i}c_{i}b_{j}d_{j}-\left(\textstyle \sum \limits _{1\leq i<j\leq n}+\sum \limits _{1\leq j<i\leq n}+\sum \limits _{1\leq i=j\leq n}\right)a_{i}d_{i}b_{j}c_{j}\\&=\textstyle \sum \limits _{1\leq i\leq n \atop 1\leq j\leq n}a_{i}c_{i}b_{j}d_{j}-\sum \limits _{1\leq i\leq n \atop 1\leq j\leq n}a_{i}d_{i}b_{j}c_{j}\\&=\left(\textstyle \sum \limits _{i=1}^{n}a_{i}c_{i}\right)\left(\textstyle \sum \limits _{j=1}^{n}b_{j}d_{j}\right)-\left(\textstyle \sum \limits _{i=1}^{n}a_{i}d_{i}\right)\left(\textstyle \sum \limits _{j=1}^{n}b_{j}c_{j}\right)\end{aligned}}}