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).
In a more compact vector notation, Lagrange's identity is expressed as:[3]
where a and b are n-dimensional vectors with components that are real numbers.
Explicitly, for complex numbers, Lagrange's identity can be written in the form:[4]
[5][6] 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:[7][8]
Using the definition of angle based upon the dot product (see also Cauchy–Schwarz inequality), the left-hand side is
The area of a parallelogram with sides |a| and |b| 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.
For a and b as vectors in R7, Lagrange's identity takes on the same form as in the case of R3[9]
Also the seven-dimensional cross product is not compatible with the Jacobi identity.
[9] A quaternion p is defined as the sum of a scalar t and a vector v:
The quaternions p and q are called imaginary if their scalar part is zero; equivalently, if
Lagrange's identity is just the multiplicativity of the norm of imaginary quaternions,
The vector form follows from the Binet-Cauchy identity by setting ci = ai and di = bi.
The second version follows by letting ci and di denote the complex conjugates of ai and bi, respectively, Here is also a direct proof.
[11] The expansion of the first term on the left side is: which means that the product of a column of as and a row of bs yields (a sum of elements of) a square of abs, which can be broken up into a diagonal and a pair of triangles on either side of the diagonal.
The second term on the left side of Lagrange's identity can be expanded as: which means that a symmetric square can be broken up into its diagonal and a pair of equal triangles on either side of the diagonal.
This proposal, originally presented in the context of a deformed Lorentz metric, is based on a transformation stemming from the product operation and magnitude definition in hyperbolic scator algebra.
[12] Lagrange's identity can be proved in a variety of ways.
reduces to the complex Lagrange's identity when fourth order terms, in a series expansion, are considered.
The two factors on the RHS are also written in terms of series
Substitution of these two results in the product identity give
in order to obtain the complex Lagrange's identity:
Higher order terms in the series produce novel identities.