User:Sintel/Dual Weil-Euclidean norm: Difference between revisions
Created page with "On some <math>p</math>-limit subgroup with <math>n</math> primes, define the <math>n \times n</math> Tenney weighting matrix <math>W</math>: $$ W = \begin{bmatrix} \log_2 2 &..." |
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and an induced norm <math>||x|| = \sqrt{\left\langle x,x \right\rangle}</math>, which we will call the [[Weil_Norms,_Tenney-Weil_Norms,_and_TWp_Interval_and_Tuning_Space#Weil-Euclidean_Norm|Weil-Euclidean norm]]. | and an induced norm <math>||x|| = \sqrt{\left\langle x,x \right\rangle}</math>, which we will call the [[Weil_Norms,_Tenney-Weil_Norms,_and_TWp_Interval_and_Tuning_Space#Weil-Euclidean_Norm|Weil-Euclidean norm]]. | ||
The inner product on the dual space can then be derived by simply inverting <math>G</math>, which | The inner product on the dual space can then be derived by simply inverting <math>G</math>, which gives the dual norm: | ||
$$ | $$ | ||
\left\langle \alpha, \beta \right\rangle = \alpha G^{-1} \beta^{\mathsf T} \\ | \left\langle \alpha, \beta \right\rangle = \alpha G^{-1} \beta^{\mathsf T} \\ | ||
||\alpha|| = \sqrt{\left\langle \alpha,\alpha \right\rangle} = \alpha G^{-1} \alpha^{\mathsf T} | ||\alpha|| = \sqrt{\left\langle \alpha,\alpha \right\rangle} = \alpha G^{-1} \alpha^{\mathsf T} | ||
$$ | $$ | ||