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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

Contents

( ∑ 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 )

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 R n .

The Binet–Cauchy identity and exterior algebra

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

( a ⋅ c ) ( b ⋅ d ) = ( a ⋅ d ) ( b ⋅ c ) + ( a ∧ b ) ⋅ ( c ∧ d )

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

( a ∧ b ) ⋅ ( c ∧ d ) = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c ) .

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

| a ∧ b | 2 = | a | 2 | b | 2 − | a ⋅ b | 2 .

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

1 = cos 2 ⁡ ( ϕ ) + sin 2 ⁡ ( ϕ )

where φ is the angle between the vectors.

Proof

Expanding the last term,

∑ 1 ≤ i < j ≤ n ( a i b j − a j b i ) ( c i d j − c j d i ) = ∑ 1 ≤ i < j ≤ n ( a i c i b j d j + a j c j b i d i ) + ∑ i = 1 n a i c i b i d i − ∑ 1 ≤ i < j ≤ n ( a i d i b j c j + a j d j b i c i ) − ∑ i = 1 n a i d i b i c i

where the second and fourth terms are the same and artificially added to complete the sums as follows:

= ∑ i = 1 n ∑ j = 1 n a i c i b j d j − ∑ i = 1 n ∑ j = 1 n a i d i b j c j .

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 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

det ( A B ) = ∑ S ⊂ { 1 , … , n } | S | = m det ( A S ) det ( B S ) ,

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

A = ( a 1 … a n b 1 … b n ) , B = ( c 1 d 1 ⋮ ⋮ c n d n ) .

References

Binet–Cauchy identity Wikipedia
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