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

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

and

$$\delta \ln[{\rm det} M] = \ln[{\rm det}(M+\delta M)]-\ln({\rm det} M)$$

$$= \ln\left[{{\rm det}(M+\delta M)\over{\rm det} M}\right]$$

$$= \ln[{\rm det} M^{-1}(M+\delta M)]$$

$$= \ln[{\rm det}(1+M^{-1}\delta M)]$$

$$\approx \ln[1+\mathop{\rm Tr}\nolimits (M^{-1}\delta M)]$$

$$\approx \mathop{\rm Tr}\nolimits (M^{-1}\delta M).$$

See also Contraction (Tensor)