User:Sintel/Dual Weil-Euclidean norm: Difference between revisions
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and an induced norm <math>||x|| = \sqrt{\left\langle x,x \right\rangle}</math>, which is 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 is 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 gives the dual norm: | The inner product on the dual space can then be derived by simply inverting <math>G</math> | ||
<ref>Taking the map <math>\Gamma: V\to V^{\ast}: x \mapsto \left\langle \cdot ,x \right\rangle</math>, we require <math>\left\langle \Gamma(x),\Gamma(y) \right\rangle^{\ast} = \left\langle x,y \right\rangle</math>.</ref> | |||
, which gives the dual norm: | |||
$$ | $$ | ||
\left\langle \alpha, \beta \right\rangle = \alpha G^{-1} \beta^{\mathsf T} \\ | \left\langle \alpha, \beta \right\rangle^{\ast} = \alpha G^{-1} \beta^{\mathsf T} \\ | ||
||\alpha|| = \sqrt{\left\langle \alpha,\alpha \right\rangle} = \sqrt{\alpha G^{-1} \alpha^{\mathsf T}} | ||\alpha||^{\ast} = \sqrt{\left\langle \alpha,\alpha \right\rangle^{\ast}} = \sqrt{\alpha G^{-1} \alpha^{\mathsf T}} | ||
$$ | $$ | ||
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G^{-1} = W^{-2} - \frac{1}{1+n} l^{\mathsf T}l | G^{-1} = W^{-2} - \frac{1}{1+n} l^{\mathsf T}l | ||
$$ | $$ | ||
<hr> | |||