On products of sums of series products
In algebra , the Binet–Cauchy identity , named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy , states that[1]
( ∑ 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 ) {\displaystyle \left(\sum _{i=1}^{n}a_{i}c_{i}\right)\left(\sum _{j=1}^{n}b_{j}d_{j}\right)=\left(\sum _{i=1}^{n}a_{i}d_{i}\right)\left(\sum _{j=1}^{n}b_{j}c_{j}\right)+\sum _{1\leq i<j\leq 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
Lagrange's identity , which is a stronger version of the
Cauchy–Schwarz inequality for the
Euclidean space R n {\textstyle \mathbb {R} ^{n}} . The Binet-Cauchy identity is a special case of the
Cauchy–Binet formula for matrix determinants.
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
( a ⋅ c ) ( b ⋅ d ) = ( a ⋅ d ) ( b ⋅ c ) + ( a ∧ b ) ⋅ ( c ∧ d ) {\displaystyle (a\cdot c)(b\cdot d)=(a\cdot d)(b\cdot c)+(a\wedge b)\cdot (c\wedge 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 ) , {\displaystyle (a\wedge b)\cdot (c\wedge d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)\,,} which can be written as
( a × b ) ⋅ ( c × d ) = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c ) {\displaystyle (a\times b)\cdot (c\times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)} in the
n = 3 case.
In the special case a = c and b = d , the formula yields
| a ∧ b | 2 = | a | 2 | b | 2 − | a ⋅ b | 2 . {\displaystyle |a\wedge b|^{2}=|a|^{2}|b|^{2}-|a\cdot b|^{2}.} When both a and b are unit vectors, we obtain the usual relation
sin 2 ϕ = 1 − cos 2 ϕ {\displaystyle \sin ^{2}\phi =1-\cos ^{2}\phi } 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 [ edit ] A relationship between the Levi–Cevita symbols and the generalized Kronecker delta is
1 k ! ε λ 1 ⋯ λ k μ k + 1 ⋯ μ n ε λ 1 ⋯ λ k ν k + 1 ⋯ ν n = δ ν k + 1 ⋯ ν n μ k + 1 ⋯ μ n . {\displaystyle {\frac {1}{k!}}\varepsilon ^{\lambda _{1}\cdots \lambda _{k}\mu _{k+1}\cdots \mu _{n}}\varepsilon _{\lambda _{1}\cdots \lambda _{k}\nu _{k+1}\cdots \nu _{n}}=\delta _{\nu _{k+1}\cdots \nu _{n}}^{\mu _{k+1}\cdots \mu _{n}}\,.} The ( a ∧ b ) ⋅ ( c ∧ d ) = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c ) {\displaystyle (a\wedge b)\cdot (c\wedge d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)} form of the Binet–Cauchy identity can be written as
1 ( n − 2 ) ! ( ε μ 1 ⋯ μ n − 2 α β a α b β ) ( ε μ 1 ⋯ μ n − 2 γ δ c γ d δ ) = δ γ δ α β a α b β c γ d δ . {\displaystyle {\frac {1}{(n-2)!}}\left(\varepsilon ^{\mu _{1}\cdots \mu _{n-2}\alpha \beta }~a_{\alpha }~b_{\beta }\right)\left(\varepsilon _{\mu _{1}\cdots \mu _{n-2}\gamma \delta }~c^{\gamma }~d^{\delta }\right)=\delta _{\gamma \delta }^{\alpha \beta }~a_{\alpha }~b_{\beta }~c^{\gamma }~d^{\delta }\,.} 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 {\displaystyle {\begin{aligned}&\sum _{1\leq i<j\leq n}(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})\\={}&{}\sum _{1\leq i<j\leq n}(a_{i}c_{i}b_{j}d_{j}+a_{j}c_{j}b_{i}d_{i})+\sum _{i=1}^{n}a_{i}c_{i}b_{i}d_{i}-\sum _{1\leq i<j\leq n}(a_{i}d_{i}b_{j}c_{j}+a_{j}d_{j}b_{i}c_{i})-\sum _{i=1}^{n}a_{i}d_{i}b_{i}c_{i}\end{aligned}}} 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 . {\displaystyle =\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i}c_{i}b_{j}d_{j}-\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i}d_{i}b_{j}c_{j}.} 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
det ( A B ) = ∑ S ⊂ { 1 , … , n } | S | = m det ( A S ) det ( B S ) , {\displaystyle \det(AB)=\sum _{S\subset \{1,\ldots ,n\} \atop |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 ) . {\displaystyle A={\begin{pmatrix}a_{1}&\dots &a_{n}\\b_{1}&\dots &b_{n}\end{pmatrix}},\quad B={\begin{pmatrix}c_{1}&d_{1}\\\vdots &\vdots \\c_{n}&d_{n}\end{pmatrix}}.} References [ edit ] Aitken, Alexander Craig (1944), Determinants and Matrices , Oliver and Boyd Harville, David A. (2008), Matrix Algebra from a Statistician's Perspective , Springer