In
algebra, Lagrange's identity, named after
Joseph Louis Lagrange, is:
which applies to any two sets and of
real or
complex numbers (or more generally, elements of a
commutative ring
In mathematics, a commutative ring is a 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 properties that are not sp ...
). This identity is a generalisation of the
Brahmagupta–Fibonacci identity and a special form of the
Binet–Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that
\left(\sum_^n a_i c_i\right)
\left(\sum_^n b_j d_j\right) =
\left(\sum_^n a_i d_i\right)
\left(\sum_^n b_j c_j\right)
+ \su ...
.
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
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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
.
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 disti ...
real coordinate space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
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, and is 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 converti ...
, 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 R
3 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 is ...
and
dot product:
Using the definition of angle based upon the
dot product (see also
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
), 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 R
7, Lagrange's identity takes on the same form as in the case of R
3[ See particularl]
§ 7.4 Cross products in R7
p. 96.
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
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. Hamilton defined a quatern ...
''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.[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:
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.
Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
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.
Most derivations use the identity as a starting point and prove in one way or another that the equality is true. In the present approach, Lagrange's identity is actually derived without assuming it ''a priori''.
Let be complex numbers and the overbar represents complex conjugate.
The product identity 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 can be expanded in terms of sums as
where means terms with order three or higher in .
The two factors on the RHS are also written in terms of series
The product of this expression up to fourth order is
Substitution of these two results in the product identity give
The product of two conjugates series can be expressed as series involving the product of conjugate terms. The conjugate series product is thus
The terms of the last two series on the LHS are grouped as
in order to obtain the complex Lagrange's identity:
In terms of the moduli,
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.
See also
* Brahmagupta–Fibonacci identity
* Lagrange's identity (boundary value problem)
*Binet–Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that
\left(\sum_^n a_i c_i\right)
\left(\sum_^n b_j d_j\right) =
\left(\sum_^n a_i d_i\right)
\left(\sum_^n b_j c_j\right)
+ \su ...
References
External links
*
{{Joseph-Louis Lagrange
Mathematical identities
Multilinear algebra
Articles containing proofs