Diferencia entre revisiones de «Norm of product»

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The product identity used as a starting point here, is a consequence
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>
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.
equality for scator algebras. However, care should be taken to avoid the divisors of zero.


The fourth order identity gives [[Lagrange identity|Lagrange's identity]].
The fourth order identity gives [[Lagrange identity|Lagrange's identity]].The sixth order identities are derived here. An extended version of these results are available in an open source journal <ref> M. Fernández-Guasti. Lagrange's identity obtained from product identity, Int. Math. Forum, 70(52):2555-2559, 2012. [http://www.m-hikari.com/imf/imf-2012/49-52-2012/fernandezguastiIMF49-52-2012.pdf]</ref>.


== sixth order 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  
 
&ensp;&ensp; <center><math>\prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=\prod_{i=1}^{n}\left(1-a_{i}^{2}\right)\prod_{i=1}^{n}\left(1-b_{i}^{2}\right)</math></center>
The non trivial identities for real numbers obtained to sixth order
series expansion of the product identity <math>\prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=\prod_{i=1}^{n}\left(1-a_{i}^{2}\right)\prod_{i=1}^{n}\left(1-b_{i}^{2}\right)</math>
are
are


<math>
<center><math>
\sum_{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)\right]+\sum_{i<j<k}^{n}\left[a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right]=\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)
\sum_{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)\right]+\sum_{i<j<k}^{n}\left[a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right]=\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)
</math>
</math></center>


and its counterpart, obtained by interchanging the variables <math>a</math> and <math>b</math>.
and its counterpart, obtained by interchanging the variables <math>a</math> and <math>b</math>. To prove it, expand the product identity in series up to sixth order. The LHS is


Expand the product identity in series up to sixth order. The LHS is
<math>
<math>
\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)
\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)
Línea 35: Línea 32:
+\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+}.
+\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+}.
</math>
</math>
Consider only the sixth order terms
Consider only the sixth order terms
<math>
<math>
Línea 40: Línea 38:
-\sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)-\sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)
-\sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)-\sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)
</math>
</math>
The RHS of the product identity is similarly expanded in series up
 
to sixth order  
The RHS of the product identity is similarly expanded in series up to sixth order  


<math>
<math>
Línea 47: Línea 45:
\left(1-\sum_{i=1}^{n}b_{i}^{2}+\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}+\mathcal{O}^{7+}\right),
\left(1-\sum_{i=1}^{n}b_{i}^{2}+\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}+\mathcal{O}^{7+}\right),
</math>
</math>
and only sixth order terms retained  
 
and only sixth order terms retained
 
<math>
<math>
\mathcal{O}^{6}\left(\textrm{RHS}\right)=-\sum_{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}-\left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right)-\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right).
\mathcal{O}^{6}\left(\textrm{RHS}\right)=-\sum_{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}-\left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right)-\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right).
</math>
</math>
These two results are equated for equal powers of <math>a^{n}b^{m}</math>. The
These two results are equated for equal powers of <math>a^{n}b^{m}</math>. The
terms <math>a^{6}</math> and <math>b^{6}</math> give trivial identities whereas the terms
terms <math>a^{6}</math> and <math>b^{6}</math> give trivial identities whereas the terms
Línea 56: Línea 57:
order identities
order identities


<math>
<center><math>
\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)
\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)
</math>
</math></center>
and
<center><math>
\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right).
</math></center>
 
----
 
<references/>


<math>
[[categoría:matematicas]]
\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right).
</math>

Revisión actual - 13:57 27 oct 2018

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. However, care should be taken to avoid the divisors of zero.

The fourth order identity gives Lagrange's identity.The sixth order identities are derived here. An extended version of these results are available in an open source journal [4].

sixth order identity

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 . To prove it, 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

and


  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.
  4. M. Fernández-Guasti. Lagrange's identity obtained from product identity, Int. Math. Forum, 70(52):2555-2559, 2012. [1]