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:<math>g_{ij}(\beta) = \frac{\partial^2}{\partial \beta^i\partial \beta^j} \left(-\log Z(\beta)\right) =
\langle \left(H_i-\langle H_i\rangle\right)\left( H_j-\langle H_j\rangle\right)\rangle</math>
This matrix is positive semi-definite, and may be interpreted as a [[metric tensor]], specifically, a [[Riemannian metric]]. Equipping the space of lagrange multipliers with a metric in this way turns it into a [[Riemannian manifold]].<ref>{{cite journal |first=Gavin E. |last=Crooks |year=2007 |title=Measuring Thermodynamic Length |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=99 |issue=10 |pages=100602 |doi=10.1103/PhysRevLett.99.100602 |arxiv=0706.0559 |bibcode=2007PhRvL..99j0602C }}</ref> The study of such manifolds is referred to as [[information geometry]]; the metric above is the [[Fisher information metric]]. Here, <math>\beta</math> serves as a coordinate on the manifold. It is interesting to compare the above definition to the simpler [[Fisher information]], from which it is inspired.
That the above defines the Fisher information metric can be readily seen by explicitly substituting for the expectation value:
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