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 Binet-Cauchy Identity
Filter Results1 entry
Connections
Has Statement Form
In Branch
Source Binet-Cauchy identity, Wikipedia
Principal Result Of
Source Binet-Cauchy identity, Wikipedia
Proved By
Source Binet-Cauchy identity, Wikipedia
Sources
1. Binet-Cauchy identity, Wikipedia
Lead paragraph, statement of the identity
for every choice of real or complex numbers (or more generally, elements of a commutative ring).
Principal Result Of: Augustin-Louis Cauchy, Lead sentence of the article on the result
In algebra, the Binet-Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that ( ∑ 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 ) + ∑ 1 ≤ i < j ≤ n ( a i b j − a j b i ) ( c i d j − c j d i ) {\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).
- In Branch: Algebra, Lead sentence
In Group: Polynomials and Elementary Algebra, Lead section
In algebra, the Binet-Cauchy identity, named after Jacques
Proved By: Augustin-Louis Cauchy, Lead
In algebra, the Binet-Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.