Binet–Cauchy identity

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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 the Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space .

The Binet–Cauchy identity and exterior algebra[edit]

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

In the special case of unit vectors a=c and b=d, the formula yields

When both vectors are unit vectors, we obtain the usual relation

where φ is the angle between the vectors.

Proof[edit]

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.

Generalization[edit]

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

In-line notes and references[edit]

  1. ^ Eric W. Weisstein (2003). "Binet-Cauchy identity". CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 228. ISBN 1-58488-347-2.