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Concept
Lagrange's identity
Summary
In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:
which applies to any two sets {a1, a2, ..., an} and {b1, b2, ..., bn} of real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity.
In a more compact vector notation, Lagrange's identity is expressed as:
where a and b are n-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form:
involving the absolute value.
Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space Rn and its complex counterpart Cn.
Geometrically, the identity asserts that the square of the volume of the parallelepiped spanned by a set of vectors is the Gram determinant of the vectors.
In terms of the wedge product, Lagrange's identity can be written
Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the parallelogram they define, in terms of the dot products of the two vectors, as
In three dimensions, Lagrange's identity asserts that if a and b are vectors in R3 with lengths |a| and |b|, then Lagrange's identity can be written in terms of the cross product and dot product:
Using the definition of angle based upon the dot product (see also Cauchy–Schwarz inequality), the left-hand side is
where θ is the angle formed by the vectors a and b. The area of a parallelogram with sides and and angle θ is known in elementary geometry to be
so the left-hand side of Lagrange's identity is the squared area of the parallelogram. The cross product appearing on the right-hand side is defined by
which is a vector whose components are equal in magnitude to the areas of the projections of the parallelogram onto the yz, zx, and xy planes, respectively.
Official source
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