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.
Lagrange's identity and exterior algebra
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
Lagrange's identity and vector calculus
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 is the angle formed by the vectors a and b. The area of a parallelogram with sides and and angle 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.
Seven dimensions
For a and b as vectors in R7, Lagrange's identity takes on the same form as in the case of R3
However, the cross product in 7 dimensions does not share all the properties of the cross product in 3 dimensions. For example, the direction of a × b in 7-dimensions may be the same as c × d even though c and d are linearly independent of a and b. Also the seven-dimensional cross product is not compatible with the Jacobi identity.
Quaternions
A quaternion p is defined as the sum of a scalar t and a vector v:
The product of two quaternions and is defined by
The quaternionic conjugate of q is defined by
and the norm squared is
The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q:
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,
since, by definition,
Proof of algebraic form
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. 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.
To expand the summation on the right side of Lagrange's identity, first expand the square within the summation:
Distribute the summation on the right side,
Now exchange the indices i and j of the second term on the right side, and permute the b factors of the third term, yielding:
Back to the left side of Lagrange's identity: it has two terms, given in expanded form by Equations () and (). The first term on the right side of Equation () ends up canceling out the first term on the right side of Equation (), yielding
which is the same as Equation (), so Lagrange's identity is indeed an identity, Q.E.D.