The linearization of a differentiable map 
at a point 
is its first-order linear
approximation
 |
where 
is the Jacobian matrix. The affine
map 
is also called the local linearization of 
.
For an autonomous system 
with an equilibrium
point 
,
writing 
gives the linearized system
 |
The eigenvalues of this matrix often determine the local stability of the equilibrium; when an eigenvalue has zero real part, the linearization alone may be inconclusive. More broadly, a linearization can also be a substitution that converts a nonlinear equation or system into a linear one.
