“Binet–Cauchy identity”的意思、由来-开放百科全书

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Binet–Cauchy identity

释义

The Binet–Cauchy identity and exterior algebra
Einstein notation
Proof
Generalization
In-line notes and references

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 a_i = c_i and b_j = d_j, 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

When {{math|1=n = 3}}, the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in {{math|n}} dimensions these become the magnitudes of the dot and wedge products. We may write it

where {{math|a}}, {{math|b}}, {{math|c}}, and {{math|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 {{math|1=n = 3}} case.

In the special case {{math|1=a = c}} and {{math|1=b = d}}, the formula yields

When both {{math|a}} and {{math|b}} are unit vectors, we obtain the usual relation

where {{math|φ}} is the angle between the vectors.

Einstein notation

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

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

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 A_S for the m×m matrix whose columns are those columns of A that have indices from S. Similarly, we write B_S 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

1. ^{{cite book |title=CRC concise encyclopedia of mathematics |author=Eric W. Weisstein |page=228 |url=https://books.google.com/books?id=8LmCzWQYh_UC&pg=PA228 |chapter=Binet-Cauchy identity |isbn=1-58488-347-2 |year=2003 |edition=2nd |publisher=CRC Press}}

3 : Mathematical identities|Multilinear algebra|Articles containing proofs

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