In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that[1]
for every choice of real or complex numbers (or more generally, elements of a commutative ring).
Setting ai = ci and bj = dj, it gives Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space
The Binet-Cauchy identity is a special case of the Cauchy–Binet formula for matrix determinants.
When n = 3, the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in n dimensions these become the magnitudes of the dot and wedge products.
We may write it
where a, b, c, and d are vectors.
It may also be written as a formula giving the dot product of two wedge products, as
which can be written as
In the special case a = c and b = d, the formula yields
When both a and b are unit vectors, we obtain the usual relation
where φ is the angle between the vectors.
This is a special case of the Inner product on the exterior algebra of a vector space, which is defined on wedge-decomposable elements as the Gram determinant of their components.
A relationship between the Levi–Cevita symbols and the generalized Kronecker delta is
δ
form of the Binet–Cauchy identity can be written as
α β
γ δ
δ
δ
γ δ
α β
δ
Expanding the last term,
where the second and fourth terms are the same and artificially added to complete the sums as follows:
This completes the proof after factoring out the terms indexed by i.
A general form, also known as the Cauchy–Binet formula, states the following: Suppose A is an m×n matrix and B is an n×m matrix.
If S is a subset of {1, ..., n} with m elements, we write AS for the m×m matrix whose columns are those columns of A that have indices from S. Similarly, we write BS for the m×m matrix whose rows are those rows of B that have indices from S. Then the determinant of the matrix product of A and B satisfies the identity
where the sum extends over all possible subsets S of {1, ..., n} with m elements.
We get the original identity as special case by setting