Editing Covariance matrix (section)

===Use in optimization===
The [[evolution strategy]], a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the [[evolution strategy]]'s covariance matrix adapts to the inverse of the [[Hessian matrix]] of the search landscape, [[up to]] a scalar factor and small random fluctuations (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation).<ref>{{cite journal | doi = 10.1016/j.tcs.2019.09.002 | first = O.M. | last = Shir | author2 = A. Yehudayoff | title = On the covariance-Hessian relation in evolution strategies | journal = Theoretical Computer Science | volume = 801 | pages = 157–174 | publisher = Elsevier | year = 2020 | doi-access = free | arxiv = 1806.03674 }}</ref>
Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.