The Binet-Cauchy identity, named for Jacques Philippe Marie Binet and Augustin-Louis Cauchy, is an algebraic identity relating the product of two sums, each built from paired terms of four sequences of numbers, to a combination of cross products and cross differences drawn from those same sequences. Setting the two pairs of sequences equal to one another turns it into Lagrange's identity, a stronger form of the Cauchy-Schwarz inequality for Euclidean space, and the identity itself is a special case of the more general Cauchy-Binet formula for matrix determinants.
Facts
StatementFor real or complex numbers, or more generally elements of a commutative ring, a_i, b_i, c_i, d_i with i from 1 to n: (sum of a_i c_i)(sum of b_j d_j) = (sum of a_i d_i)(sum of b_j c_j) + the sum over 1 <= i < j <= n of (a_i b_j - a_j b_i)(c_i d_j - c_j d_i). 1 Classification
Statement Form Statement Form Sources
1. Binet-Cauchy identity, Wikipedia
Lead paragraph, statement of the identityQuote, Lead paragraph, statement of the identity
for every choice of real or complex numbers (or more generally, elements of a commutative ring).
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