Editing Jacobian matrix and determinant (section)

== Other uses ==

=== Dynamical systems ===

Consider a [[dynamical system]] of the form <math>\dot{\mathbf{x}} = F(\mathbf{x})</math>, where <math>\dot{\mathbf{x}}</math> is the (component-wise) derivative of <math>\mathbf{x}</math> with respect to the [[evolution parameter]] <math>t</math> (time), and <math>F \colon \mathbb{R}^{n} \to \mathbb{R}^{n}</math> is differentiable. If <math>F(\mathbf{x}_{0}) = 0</math>, then <math>\mathbf{x}_{0}</math> is a [[stationary point]] (also called a [[steady state]]). By the [[Hartman–Grobman theorem]], the behavior of the system near a stationary point is related to the [[eigenvalue]]s of <math>\mathbf{J}_{F} \left( \mathbf{x}_{0} \right)</math>, the Jacobian of <math>F</math> at the stationary point.<ref>{{cite book |first1=D. K. |last1=Arrowsmith |first2=C. M. |last2=Place |title=Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour |chapter=The Linearization Theorem |publisher=Chapman & Hall |location=London |year=1992 |isbn=0-412-39080-9 |pages=77–81 |chapter-url=https://books.google.com/books?id=8qCcP7KNaZ0C&pg=PA77 }} </ref> Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.<ref>{{cite book |first1=Morris |last1=Hirsch |first2=Stephen |last2=Smale |title=Differential Equations, Dynamical Systems and Linear Algebra |year=1974 |isbn=0-12-349550-4 }}</ref>

=== Newton's method ===

A square system of coupled nonlinear equations can be solved iteratively by [[Newton's method#Systems of equations|Newton's method]]. This method uses the Jacobian matrix of the system of equations.

===Regression and least squares fitting===
The Jacobian serves as a linearized [[design matrix]] in statistical [[regression analysis|regression]] and [[curve fitting]]; see [[non-linear least squares]]. The Jacobian is also used in random matrices, moments, local sensitivity and statistical diagnostics.<ref>{{cite journal|last1=Liu|first1=Shuangzhe |last2=Leiva|first2=Victor|last3=Zhuang|first3=Dan|last4=Ma|first4=Tiefeng | last5=Figueroa-Zúñiga | first5=Jorge I.|date=March 2022|title=Matrix differential calculus with applications in the multivariate linear model and its diagnostics|journal=Journal of Multivariate Analysis |volume=188|pages=104849|doi=10.1016/j.jmva.2021.104849|doi-access=free}}</ref><ref>{{Cite journal| last1=Liu|first1=Shuangzhe|
last2= Trenkler|first2=Götz| last3=Kollo|first3=Tõnu| 
last4=von Rosen|first4=Dietrich| 
last5=Baksalary|first5=Oskar Maria| 
date= 2023|
title= Professor Heinz Neudecker and matrix differential calculus| 
journal= Statistical Papers |volume=65 |issue=4 |pages=2605–2639 | language=en | doi= 10.1007/s00362-023-01499-w|s2cid=263661094 }}</ref>