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==Norm of a product==


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 <math>\mathbb{R}^{n}</math>
are possible <ref>P. Fjelstad and S. G. Gal., ''n-dimensional hyperbolic complex numbers'', Adv. Appl. Clifford Alg., 8(1), 1998, p. 47–68</ref>, <ref>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</ref>.
One 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 <ref> 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.</ref>.
The product identity used as a starting point here, is a consequence
of the <math>\left\Vert \mathbf{ab}\right\Vert =\left\Vert \mathbf{a}\right\Vert \left\Vert \mathbf{b}\right\Vert </math>
equality for scator algebras.
The fourth order identity gives [[Lagrange identity|Lagrange's identity]].


== sixth order identity ==
== sixth order identity ==

Revisión del 13:37 19 ago 2012

Norm of a product

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 [1], [2]. One 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.

The fourth order identity gives Lagrange's identity.

sixth order identity

Sixth and higher order terms produce other series 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

  1. P. Fjelstad and S. G. Gal., n-dimensional hyperbolic complex numbers, Adv. Appl. Clifford Alg., 8(1), 1998, p. 47–68
  2. 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
  3. 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.