In the

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, Lagrange's identity, named after

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia\begin \left( \sum_^n a_k^2\right) \left(\sum_^n b_k^2\right) - \left(\sum_^n a_k b_k\right)^2 & = \sum_^ \sum_^n \left(a_i b_j - a_j b_i\right)^2 \\ & \left(= \frac \sum_^n \sum_^n (a_i b_j - a_j b_i)^2\right), \end which applies to any two sets and of real or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s (or more generally, elements of a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

). 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:

\left\,  \mathbf a \right\, ^2 \left\,  \mathbf b \right\, ^2 - (\mathbf \cdot \mathbf )^2 = \sum_ \left(a_i b_j - a_j b_i \right)^2 \, ,

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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...

as an

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form:

\left( \sum_^n , a_k, ^2\right) \left(\sum_^n , b_k, ^2\right) - \left, \sum_^n a_k b_k\^2 = \sum_^ \sum_^n \left, a_i \overline_j - a_j \overline_i\^2

involving the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

. Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

real coordinate space In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''. ...

R^''n'' and its complex counterpart C^''n''. Geometrically, the identity asserts that the square of the volume of the parallelepiped spanned by a set of vectors is the

Gram determinant In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...

of the vectors.

Lagrange's identity and exterior algebra

In terms of the

wedge product A wedge is a triangular shaped tool, a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a fo ...

, Lagrange's identity can be written

(a \cdot a)(b \cdot b) - (a \cdot b)^2 = (a \wedge b) \cdot (a \wedge b).

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

\, a \wedge b\,  = \sqrt = \sqrt.

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 In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

and

, \mathbf, ^2 , \mathbf, ^2 - (\mathbf  \cdot \mathbf)^2 = , \mathbf a \times \mathbf b, ^2

Using the definition of angle based upon the

(see also

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

), the left-hand side is

\left, \mathbf\^2 \left, \mathbf\^2 \left( 1- \cos^2\theta\right) = \left, \mathbf\^2 \left, \mathbf\^2\sin^2\theta

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

\left, \mathbf\ \left, \mathbf\ \left, \sin\theta\,

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

\mathbf\times\mathbf = \left(a_2 b_3- a_3 b_2\right)\mathbf + \left(a_3 b_1 - a_1 b_3\right)\mathbf + \left(a_1 b_2 - a_2 b_1\right)\mathbf

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 See particularl
§ 7.4 Cross products in R⁷
p. 96.

, \mathbf, ^2 , \mathbf, ^2 -, \mathbf \cdot \mathbf, ^2 = , \mathbf \times \mathbf, ^2 \ ,

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 In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven-dime ...

is not compatible with the

Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...

Quaternions

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

''p'' is defined as the sum of a scalar ''t'' and a vector v:

p = t + \mathbf v = t + x \ \mathbf  i +y \ \mathbf j + z\  \mathbf k.

The product of two quaternions and is defined by

pq = (st - \mathbf\cdot\mathbf) + s \ \mathbf + t \ \mathbf  + \mathbf\times\mathbf.

The quaternionic conjugate of ''q'' is defined by

\overline = t - \mathbf,

and the norm squared is

, q, ^2 = q\overline = t^2 \ + \ x ^2   + \ y^2 \ +\  z^2.

The multiplicativity of the norm in the quaternion algebra provides, for quaternions ''p'' and ''q'':

\left, pq\ = \left, p\ \left, q\.

The quaternions ''p'' and ''q'' are called imaginary if their scalar part is zero; equivalently, if

p = \mathbf,\quad q = \mathbf.

Lagrange's identity is just the multiplicativity of the norm of imaginary quaternions,

, \mathbf\mathbf, ^2 = , \mathbf, ^2, \mathbf, ^2,

since, by definition,

, \mathbf\mathbf, ^2 = (\mathbf\cdot\mathbf)^2 + , \mathbf\times\mathbf, ^2.

Proof of algebraic form

The vector form follows from the Binet-Cauchy identity by setting ''c_i'' = ''a_i'' and ''d_i'' = ''b_i''. The second version follows by letting ''c_i'' and ''d_i'' denote the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

s of ''a_i'' and ''b_i'', respectively, Here is also a direct proof.See, for example
Frank Jones, Rice University
page 4 in Chapter 7 of
book still to be published
The expansion of the first term on the left side is: which means that the product of a column of ''a''s and a row of ''b''s yields (a sum of elements of) a square of ''ab''s, 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:

\sum_^ \sum_^n (a_i b_j - a_j b_i)^2 = \sum_^ \sum_^n \left(a_i^2 b_j^2 + a_j^2 b_i^2 - 2 a_i b_j a_j b_i\right).

Distribute the summation on the right side,

\sum_^ \sum_^n (a_i b_j - a_j b_i)^2 = \sum_^ \sum_^n a_i^2 b_j^2 + \sum_^ \sum_^n a_j^2 b_i^2 - 2 \sum_^ \sum_^n a_i b_j a_j b_i .

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. Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...

Proof of Lagrange's identity for complex numbers

Normed division algebras require that the norm of the product is equal to the product of the norms. Lagrange's identity exhibits this equality. The product identity used as a starting point here, is a consequence of the norm of the product equality with the product of the norm for scator algebras. 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.M. Fernández-Guasti, ''Alternative realization for the composition of relativistic velocities'', Optics and Photonics 2011, vol. 8121 of The nature of light: What are photons? IV, pp. 812108–1–11. SPIE, 2011. Lagrange's identity can be proved in a variety of ways. Let

a_,b_\in\mathbb

be complex numbers and the overbar represents complex conjugate. The product identity

\prod_^ \left(1 - a_ \bar_ - b_ \bar_ + a_ \bar_ b_ \bar_\right) = \prod_^ \left(1 - a_ \bar_\right) \prod_^ \left(1 - b_ \bar_\right)

reduces to the complex Lagrange's identity when fourth order terms, in a series expansion, are considered. In order to prove it, expand the product on the LHS of the product identity in terms of series up to fourth order. To this end, recall that products of the form

\left(1 +x_\right)

can be expanded in terms of sums as

\prod_^\left(1+x_\right) = 1+\sum_^x_+\sum_^x_x_+\mathcal^(x),

where

\mathcal^(x)

means terms with order three or higher in

x

\prod_^\left(1 - a_ \bar_ - b_ \bar_ + a_ \bar_ b_ \bar_\right) = 1-\sum_^ \left(a_\bar_ + b_ \bar_\right) + \sum_^ a_ \bar_ b_ \bar_
+ \sum_^ \left(a_ \bar_ a_ \bar_ + b_ \bar_ b_ \bar_\right)+\sum_^\left(a_ \bar_ b_ \bar_ + a_ \bar_ b_ \bar_\right) + \mathcal^.

The two factors on the RHS are also written in terms of series

\prod_^\left(1-a_\bar_\right)\prod_^\left(1-b_\bar_\right)=\left(1-\sum_^a_\bar_+\sum_^a_\bar_a_\bar_+\mathcal^\right)
\left(1-\sum_^b_\bar_+\sum_^b_\bar_b_\bar_+\mathcal^\right).

The product of this expression up to fourth order is

\prod_^\left(1-a_\bar_\right)\prod_^\left(1-b_\bar_\right)=1-\sum_^\left(a_\bar_+b_\bar_\right)
+\left(\sum_^a_\bar_\right)\left(\sum_^b_\bar_\right)+\sum_^\left(a_\bar_a_\bar_+b_\bar_b_\bar_\right)+\mathcal^.

Substitution of these two results in the product identity give

\sum_^a_\bar_b_\bar_+\sum_^\left(a_\bar_b_\bar_+a_\bar_b_\bar_\right)=\left(\sum_^a_\bar_\right)\left(\sum_^b_\bar_\right).

The product of two conjugates series can be expressed as series involving the product of conjugate terms. The conjugate series product is

\left(\sum_^ x_\right)\left(\sum_^\bar_\right) = \sum_^ x_\bar_+\sum_^\left(x_\bar_+\bar_x_\right),

thus

\left(\sum_^a_b_\right)\left(\sum_^\overline\right)-\sum_^\left(a_b_\bar_\bar_+\bar_\bar_a_b_\right)+\sum_^\left(a_\bar_b_\bar_+a_\bar_b_\bar_\right)
=\left(\sum_^a_\bar_\right)\left(\sum_^b_\bar_\right).

The terms of the last two series on the LHS are grouped as

a_\bar_b_\bar_+a_\bar_b_\bar_-a_b_\bar_\bar_-\bar_\bar_a_b_ = \left(a_\bar_-a_\bar_\right)\left(\bar_b_-\bar_b_\right),

in order to obtain the complex Lagrange's identity:

\left(\sum_^a_b_\right)\left(\sum_^\overline\right)+\sum_^\left(a_\bar_-a_\bar_\right)\left(\overline\right)=\left(\sum_^a_\bar_\right)\left(\sum_^b_\bar_\right).

In terms of the moduli,

\left, \sum_^a_b_\^+\sum_^\left, a_\bar_-a_\bar_\^=\left(\sum_^\left, a_\^\right)\left(\sum_^\left, b_\^\right).

Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce novel identities.

References

External links

* {{Joseph-Louis Lagrange Algebraic identities Multilinear algebra Articles containing proofs