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There is a famous equation called Binet-Cauchy identity which states that $$ \left(\sum_{i=1}^n a_i c_i\right) \left(\sum_{j=1}^n b_j d_j\right) = \left(\sum_{i=1}^n a_i d_i\right) \left(\sum_{j=1}^n b_j c_j\right) + \sum_{1 \le i \lt j \le n} \left(a_i b_j - a_j b_i\right)\left(c_i d_j - c_j d_i\right) $$

I understood the proof. But the problem arises when I think why someone has to bother about this on the first place ?

So my questions are:

  • What it tells us about ?

  • Why someone will go from something like on the left side of the equation to something like on the right side of the equation as the left side has the same information and it's more compact ?

  • What are the applications of this equation ?

Abhisek
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  • Chapter $3$ of The Cauchy-Schwarz Master Class by J. Michael Steele might be interesting for you. The Cauchy-Binet identity appears as exercise $3.7$ and the author gives some discussion. It helps to see it as a generalization of Lagrange's identity. – Banach Manifold Sep 03 '16 at 11:26
  • It is a particular case of the Cauchy-Binet formula for determinants: https://en.wikipedia.org/wiki/Binet%E2%80%93Cauchy_identity#Generalization – darij grinberg Sep 08 '16 at 22:39

1 Answers1

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This can be expressed as a problem in clifford algebra--an algebra which includes both inner products and wedge products. One way of denoting this equality would be, for vectors $a, b, c, d \in \mathbb K^n$ for some commutative ring $K$,

$$(a \cdot d)(b \cdot c) - (a \cdot d)(b \cdot d) = a \cdot [b \cdot (c \wedge d)] = (a \wedge b) \cdot (c \wedge d)$$

These expressions can be understood geometrically. I will consider only the case in which these vectors are unit, as their magnitudes will only change the answer by a scalar multiple. Doing so, the expressions can be understood as:

1) For the middle expression: "From the plane formed by the vectors $c, d$, take the unique vector in that plane that is also perpendicular to $b$, and find its orthogonal projection along the vector $a$."

2) For the final expression: "Take the plane formed by the vectors $c, d$ and project it onto the plane formed by $a, b$."

The theorem says these two expressions are equal.

Muphrid
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