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Línea 5: |
Línea 5: |
| Normed division algebras require that the norm of the product is equal | | Normed division algebras require that the norm of the product is equal |
| to the product of the norms. Lagrange's identity exhibits this equality. | | to the product of the norms. Lagrange's identity exhibits this equality. |
| Due to Hurwitz theorem, it admits this interpretation only for algebras
| | One approach has been presented in the context of a deformed Lorentz metric. |
| isomorphic to the real numbers, complex numbers, quaternions and octonions. If divisors of zero
| | The product identity used as a starting point here, is a consequence of the [[Norm of product|product of the norms]] equality for scator algebras. This proposal is based on a transformation stemming from the product operation and |
| are allowed, many other algebraic structures in <math>\mathbb{R}^{n}</math>
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| 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>.
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| 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>. | | 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 [[Norm of product|product of the norms]] equality for scator algebras.
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| Lagrange's identity can be proved in a variety of ways <ref name="Steele"> J. Michael Steele, ''Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities'', Adv. Appl. Clifford Alg., CUP 2004 </ref>. | | Lagrange's identity can be proved in a variety of ways <ref name="Steele"> J. Michael Steele, ''Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities'', Adv. Appl. Clifford Alg., CUP 2004 </ref>. |
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. An extended version of these results are available in an open source journal [1].
Normed division algebras require that the norm of the product is equal
to the product of the norms. Lagrange's identity exhibits this equality.
One approach has been presented in the context of a deformed Lorentz metric.
The product identity used as a starting point here, is a consequence of the product of the norms equality for scator algebras. This proposal is based on a transformation stemming from the product operation and
magnitude definition in hyperbolic scator algebra [2].
Lagrange's identity can be proved in a variety of ways [3].
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.
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 Error al representar (SVG (MathML puede ser habilitado mediante un plugin de navegador): respuesta no válida («Math extension cannot connect to Restbase.») del servidor «https://en.wikipedia.org/api/rest_v1/»:): \mathcal{O}^{3+}(x)
means terms with order three or higher
in .
The two factors on the RHS are also written in terms of series
Error al representar (SVG (MathML puede ser habilitado mediante un plugin de navegador): respuesta no válida («Math extension cannot connect to Restbase.») del servidor «https://en.wikipedia.org/api/rest_v1/»:): \prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}\right)\prod_{i=1}^{n}\left(1-b_{i}\bar{b}_{i}\right)=\left(1-\sum_{i=1}^{n}a_{i}\bar{a}_{i}+\sum_{i<j}^{n}a_{i}\bar{a}_{i}a_{j}\bar{a}_{j}+\mathcal{O}^{5+}\right) \left(1-\sum_{i=1}^{n}b_{i}\bar{b}_{i}+\sum_{i<j}^{n}b_{i}\bar{b}_{i}b_{j}\bar{b}_{j}+\mathcal{O}^{5+}\right).
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
Error al representar (SVG (MathML puede ser habilitado mediante un plugin de navegador): respuesta no válida («Math extension cannot connect to Restbase.») del servidor «https://en.wikipedia.org/api/rest_v1/»:): \prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=1+\sum_{i=1}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right) +\sum_{i<j}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right) +\sum_{i<j<k}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right)\left(-a_{k}^{2}-b_{k}^{2}+a_{k}^{2}b_{k}^{2}\right)+\mathcal{O}^{7+}.
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 Error al representar (SVG (MathML puede ser habilitado mediante un plugin de navegador): respuesta no válida («Math extension cannot connect to Restbase.») del servidor «https://en.wikipedia.org/api/rest_v1/»:): b^{6}
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 [3], this
proof is yet another way to obtain the CS inequality.
- ↑ M. Fernández-Guasti. Lagrange's identity obtained from product identity, Int. Math. Forum, 70(52):2555-2559, 2012. [1]
- ↑ 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.
- ↑ 3,0 3,1 J. Michael Steele, Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities, Adv. Appl. Clifford Alg., CUP 2004