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Trace (Tensor)

The trace of a second-Rank Tensor $T$ is a Scalar given by the Contracted mixed Tensor equal to $T_i^i$. The trace satisfies

\mathop{\rm Tr}\nolimits \left[{M^{-1}(x){\partial\over\part...
= {\partial\over\partial x^\lambda} \ln[{\rm det}(x)],

$\displaystyle \delta \ln[{\rm det} M]$ $\textstyle =$ $\displaystyle \ln[{\rm det}(M+\delta M)]-\ln({\rm det} M)$  
  $\textstyle =$ $\displaystyle \ln\left[{{\rm det}(M+\delta M)\over{\rm det} M}\right]$  
  $\textstyle =$ $\displaystyle \ln[{\rm det} M^{-1}(M+\delta M)]$  
  $\textstyle =$ $\displaystyle \ln[{\rm det}(1+M^{-1}\delta M)]$  
  $\textstyle \approx$ $\displaystyle \ln[1+\mathop{\rm Tr}\nolimits (M^{-1}\delta M)]$  
  $\textstyle \approx$ $\displaystyle \mathop{\rm Tr}\nolimits (M^{-1}\delta M).$  

See also Contraction (Tensor)

© 1996-9 Eric W. Weisstein