Binet–Cauchy identity

In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that $${\displaystyle \left(\sum _{i=1}^{n}a_{i}c_{i}\right)\left(\sum _{j=1}^{n}b_{j}d_{j}\right)=\left(\sum _{i=1}^{n}a_{i}d_{i}\right)\left(\sum _{j=1}^{n}b_{j}c_{j}\right)+\sum _{1\leq i<j\leq n}(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})}$$ for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting a_i = c_i and b_j = d_j, it gives Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space ${\textstyle \mathbb {R} ^{n}}$. The Binet-Cauchy identity is a special case of the Cauchy–Binet formula for matrix determinants.