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determinant_differentiation [2016/07/18 23:15]
nikolaj
determinant_differentiation [2016/07/21 01:00] (current)
nikolaj
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 This comes from [[http://​en.wikipedia.org/​wiki/​Jacobi%27s_formula|Jacobi'​s formula]]: This comes from [[http://​en.wikipedia.org/​wiki/​Jacobi%27s_formula|Jacobi'​s formula]]:
  
-${\mathrm d} \log\left(\det (F(t))\right\mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$+${\mathrm d} \det (F(t)) ​\det (F(t)) \mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$
  
 where $F(t)$ is a parameter dependent matrix where $F(t)$ is a parameter dependent matrix
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 This is a special case of the product rule and generalizes ​ This is a special case of the product rule and generalizes ​
  
-${\mathrm d}\left(u\cdot v\right) = u\,{\mathrm d}v+v\,{\mathrm d}u\cdot v\left(\dfrac{1}{u}{\mathrm d}u+\dfrac{1}{v}{\mathrm d}v\right)$.+${\mathrm d}\left(a\cdot b\right) = a\,{\mathrm d}b+b\,{\mathrm d}a\cdot b\left(\dfrac{1}{a}{\mathrm d}a+\dfrac{1}{b}{\mathrm d}b\right)$.
  
 which you get for which you get for
  
-$F(t) := \mathrm{diag}(u(t),v(t))$+$F(t) := \mathrm{diag}(a(t),b(t))$,
  
-The expression $\dfrac{1}{u}{\mathrm d}u$ is the so called logarithmic derivative of $u$.+which can be seen $a(t)\cdot b(t)=\det\,​F(t)$ representing the changing area of a rectangle. 
 + 
 +The expression $\dfrac{1}{a}{\mathrm d}a$ is the so called logarithmic derivative of $aand scale invariant.
  
 == Perspective == == Perspective ==
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