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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 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.

The Binet–Cauchy identity and exterior algebra

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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 n = 3 case.

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.

Einstein notation

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A relationship between the Levi–Cevita symbols and the generalized Kronecker delta is

The form of the Binet–Cauchy identity can be written as

Proof

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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

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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

Notes

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

References

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  • Aitken, Alexander Craig (1944), Determinants and Matrices, Oliver and Boyd
  • Harville, David A. (2008), Matrix Algebra from a Statistician's Perspective, Springer