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determinant_differentiation [2016/07/18 23:13] nikolaj |
determinant_differentiation [2016/07/21 00:59] 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]]: | ||
- | $\dfrac{1}{\mathrm{det}(F(t))}{\mathrm d} \det (F(t)) = \mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$ | + | ${\mathrm d} \det (F(t)) = \det (F(t)) \mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$ |
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 | ||
- | ${\mathrm d}\left(u\cdot v\right) = u\,{\mathrm d}v+v\,{\mathrm d}u = u\cdot v\left(\dfrac{1}{u}{\mathrm d}u+\dfrac{1}{v}{\mathrm d}v\right)$. | + | ${\mathrm d}\left(a\cdot b\right) = a\,{\mathrm d}b+b\,{\mathrm d}a = a\cdot b\left(\dfrac{1}{a}{\mathrm d}a+\dfrac{1}{b}{\mathrm d}b\right)$. |
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 to represent the growing area of a rectangle. |
+ | |||
+ | The expression $\dfrac{1}{a}{\mathrm d}a$ is the so called logarithmic derivative of $a$ and scale invariant. | ||
== Perspective == | == Perspective == |