Lagrange's identity obtained from product identity
We present an identity of products that reduces to Lagrange's identity
when a series expansion to fourth order terms are considered. Sixth
and higher order terms produce other series identities.
Normed division algebras require that the norm of the product is equal
to the product of the norms. Lagrange's identity exhibits this equality.
Due to Hurwitz theorem, it admits this interpretation only for algebras
isomorphic to the real numbers, complex
numbers, quaternions
and octonions . If divisors of zero
are allowed, many other algebraic structures in
are possible. Two such possibilities for hyperbolic numbers has been
introduced by Fjelstad and Gal [1]
and more recently by Catoni et al. [2].
Another approach has been presented
in the context of a deformed Lorentz metric. This latter proposal
is based on a transformation stemming from the product operation and
magnitude definition in hyperbolic scator algebra [3].
The product identity used as a starting point here, is a consequence
of the
equality for scator algebras.
Lagrange's identity can be proved in a variety of ways [4].
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 \emph{a
priori}. The pseudo-norm of the product identity used in the derivation
has the strength to imply an infinite number of identities. An example
when sixth order terms are retained is shown here. The ease of the
derivation has induced us to present it for complex numbers.
Lagrange's identity for complex numbers
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.
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 modulii,
other identities
The non trivial identities for real numbers obtained to sixth order
series expansion of the product identity
are
and its counterpart, obtained by interchanging the variables and .
Expand the product identity in series up to sixth order. The LHS is
Consider only the sixth order terms
The RHS of the product identity is similarly expanded in series up
to sixth order
and only sixth order terms retained
These two results are equated for equal powers of . The
terms and give trivial identities whereas the terms
involving and give the non trivial sixth
order identities
conclusions
Lagrange's identity for complex numbers has been obtained from a straight-forward
product identity. The procedure is elementary and very economical.
A derivation for the reals is obviously even more succinct. In a wider
context, this product identity can be seen as a consequence of the
relationship for scator algebras. Since the Cauchy–Schwarz inequality
is a particular case of Lagrange's identity [4], this
proof is yet another way to obtain the CS inequality.
- ↑ P. Fjelstad and S. G. Gal., n-dimensional hyperbolic complex numbers, Adv. Appl. Clifford Alg., 8(1), 1998, pag.47–68
- ↑ F. Catoni, R. Cannata, E. Nichelatti, and P. Zampetti, Commmutative hypercomplex numbers and functions of hypercomplex variable: a matrix study, Adv. Appl. Clifford Alg., 15(2), 2005, pag.183–212
- ↑ 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.
- ↑ 4,0 4,1 J. Michael Steele, Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities, Adv. Appl. Clifford Alg., CUP 2004