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Tensor Trace

The trace of a second-tensor rank tensor T is a scalar given by the contracted mixed tensor equal to T_i^i.

The trace satisfies

Tr[M^(-1)(x)partial/(partialx^lambda)M(x)]=partial/(partialx^lambda)lndet[M(x)].
(1)

For a small change deltaM in a given tensor M, the trace can be computed approximately as

deltaln[det(M)]=ln[det(M+deltaM)]-ln(detM)
(2)
=ln[(det(M+deltaM))/(detM)]
(3)
=ln[detM^(-1)(M+deltaM)]
(4)
=ln[det(1+M^(-1)deltaM)]
(5)
approx ln[1+Tr(M^(-1)deltaM)]
(6)
approx Tr(M^(-1)deltaM).
(7)

See also

Group Character, Matrix Trace, Tensor Contraction

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Cite this as:

Weisstein, Eric W. "Tensor Trace." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TensorTrace.html

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