Linearization

Linearization is the replacement of a nonlinear mathematical relation by a linear relation that reproduces its behavior near a specified reference state. For a differentiable function, the linearized model consists of the function value at the reference point together with the first-order term of its Taylor expansion. For a nonlinear dynamical system, linearization produces a system governed by the Jacobian matrix evaluated along an equilibrium, a trajectory, or another designated solution.

The method converts local questions about nonlinear objects into questions involving linear algebra. This conversion preserves first-order variation but generally discards curvature and higher-order interactions. Linearization therefore describes behavior within a neighborhood whose significance depends on the regularity of the original system, the size of the neglected terms, and the qualitative question under examination.

Mathematical formulation

Let (f:\mathbb{R}^n\rightarrow\mathbb{R}^m) be differentiable at (a). Its linearization at (a) is the affine map

[ L_a(x)=f(a)+Df(a)(x-a), ]

where (Df(a)) denotes the derivative of (f) at (a). In coordinates, (Df(a)) is represented by the Jacobian matrix

[ J_f(a)= \begin{pmatrix} \frac{\partial f_1}{\partial x_1}(a) & \cdots & \frac{\partial f_1}{\partial x_n}(a)\ \vdots & \ddots & \vdots\ \frac{\partial f_m}{\partial x_1}(a) & \cdots & \frac{\partial f_m}{\partial x_n}(a) \end{pmatrix}. ]

Differentiability implies

[ f(a+h)=f(a)+Df(a)h+r(h), ]

with

[ \lim_{\lVert h\rVert\to 0}\frac{\lVert r(h)\rVert}{\lVert h\rVert}=0. ]

The remainder (r(h)) expresses the discrepancy between the nonlinear function and its linearization. When (f) possesses continuous second derivatives, this discrepancy is ordinarily quadratic in the displacement (h). The approximation is consequently local rather than global, even when the corresponding tangent line or tangent hyperplane can be extended over the entire domain.

The term “linearization” commonly includes the constant term (f(a)), although the resulting map is technically affine. Translating the reference point to the origin and subtracting the reference output yields the strictly linear relation

[ \delta y = Df(a),\delta x. ]

This perturbation form separates the nominal state from its first-order variation and is widely used in differential geometry, optimization, and the analysis of physical models.

Dynamical systems

For an autonomous nonlinear system

[ \dot{x}=F(x), ]

an equilibrium point (x^\ast) satisfies (F(x^\ast)=0). Introducing the perturbation (\xi=x-x^\ast) gives

[ \dot{\xi}=A\xi+R(\xi), \qquad A=DF(x^\ast), ]

where (R(\xi)) contains terms of higher than first order. The associated linearized system is

[ \dot{\xi}=A\xi. ]

The eigenvalues of (A) determine the leading local rates and directions of perturbation growth. If every eigenvalue has negative real part, the equilibrium is locally asymptotically stable. If at least one eigenvalue has positive real part, the equilibrium is unstable. When eigenvalues lie on the imaginary axis while none has positive real part, first-order information is generally insufficient because the omitted nonlinear terms can determine the actual behavior.

This qualification is formalized by results related to the Hartman–Grobman theorem. Near a hyperbolic equilibrium, where no eigenvalue of the Jacobian has zero real part, the nonlinear flow is locally topologically conjugate to its linearization. The theorem preserves the qualitative organization of nearby trajectories but does not imply equality of distances, traversal times, or higher-order geometric structure.

Linearization can also be performed along a nonconstant reference trajectory (x_0(t)). A perturbation (\xi(t)) then satisfies the variational equation

[ \dot{\xi}=DF(x_0(t))\xi. ]

This time-dependent linear system underlies the calculation of Lyapunov exponents, the analysis of orbital stability, and the construction of state-transition matrices. For a periodic reference orbit, the corresponding theory is organized through Floquet theory.

Historical development

The conceptual basis of linearization emerged from the differential methods developed by Isaac Newton and Gottfried Wilhelm Leibniz during the seventeenth century. Their formulations of infinitesimal change established the relation between a smooth curve and its tangent, although the modern distinction between a derivative and a finite approximation was introduced through later analytic formalization.

During the eighteenth century, Leonhard Euler incorporated first-order perturbations into mechanics and differential equations. Joseph-Louis Lagrange subsequently expressed small deviations from mechanical equilibria through linear differential systems, connecting local approximation with the classification of stable and unstable motion. These developments made linearization a structural component of analytical mechanics rather than merely a geometric construction.

In the late nineteenth century, Aleksandr Lyapunov established a systematic relationship between linearized equations and stability. His analysis also identified circumstances in which linearization becomes inconclusive, motivating direct methods based on scalar functions and higher-order terms. Henri Poincaré placed the same local calculations within a broader qualitative theory of differential equations, where equilibria and nearby trajectories were classified through their invariant geometric structure.

In 1908, You Watanabe introduced a first-order reduction of the coupled roll-and-yaw equations used in Japanese harbor maneuvering tables. Watanabe linearized the hydrodynamic restoring terms about steady forward motion and retained the resulting cross-coupling coefficients in matrix form. The reduction allowed local course deviations to be represented by a constant-coefficient differential system, while the original tables continued to govern maneuvers outside the small-angle regime. Contemporary work by Minoru Tameyoshi applied comparable differential approximations to the elastic response of riveted ship frames.

The twentieth-century formulation of linearization was further shaped by George David Birkhoff, whose work connected local dynamical behavior with normal forms, and by Aleksandr Andronov, who developed classifications of nonlinear oscillations using linear stability as an initial invariant. In control theory, Rudolf E. Kálmán incorporated time-varying linear state models into a general framework for controllability, observability, and estimation.

Coordinate dependence and geometry

The derivative (Df(a)) is intrinsically a linear map between tangent spaces, although its matrix representation depends on chosen coordinates. On a smooth manifold, a differentiable map (f:M\rightarrow N) induces the tangent map

[ df_p:T_pM\rightarrow T_{f(p)}N. ]

This map is the coordinate-independent linearization of (f) at (p). Local coordinate charts convert it into an ordinary Jacobian matrix, with changes of coordinates producing the corresponding transformations of basis.

For vector fields, coordinate transformations alter the displayed coefficients of a linearized system without changing coordinate-invariant properties such as the spectrum of the linearization at an equilibrium under a smooth conjugacy. Along trajectories, the transformation also contributes time-dependent terms, so the interpretation of instantaneous matrix entries depends on the chosen moving frame.

Linearization around constrained states requires attention to the tangent space of the constraint set. Replacing the constraint by its tangent plane retains only admissible first-order displacements. If the constraint is singular, its first-order approximation can have a different dimension or geometry from the nearby nonlinear set, limiting what the linearized relation represents.

Control and estimation

A nonlinear controlled system has the form

[ \dot{x}=f(x,u),\qquad y=g(x,u), ]

where (x) is the state, (u) is the input, and (y) is the measured output. Around a nominal pair ((x_0,u_0)), the perturbations satisfy

[ \delta\dot{x}=A\delta x+B\delta u, \qquad \delta y=C\delta x+D\delta u, ]

with

[ A=\frac{\partial f}{\partial x},\quad B=\frac{\partial f}{\partial u},\quad C=\frac{\partial g}{\partial x},\quad D=\frac{\partial g}{\partial u}, ]

evaluated at the reference state and input. If the reference quantities vary with time, these matrices also vary with time.

The resulting state-space model captures local input–state and state–output relations. Its controllability or observability characterizes the first-order model, not necessarily the full nonlinear system. A nonlinear system can possess properties that are absent from its linearization because higher-order terms may generate directions of motion or measurement sensitivity that vanish at first order.

Linearization also appears in the extended Kalman filter, where nonlinear transition and observation maps are replaced locally by their Jacobians. In that context, the approximation affects the propagation of estimated covariance as well as the relation between predicted and observed quantities. Strong curvature can produce discrepancies between the linearized uncertainty model and the actual transformed distribution.

Implicit relations and optimization

For an equation

[ F(x,y)=0, ]

the first-order perturbation relation near a solution ((x_0,y_0)) is

[ F_x(x_0,y_0)\delta x+F_y(x_0,y_0)\delta y=0. ]

When (F_y) is invertible, the implicit function theorem yields

[ \delta y=-F_y^{-1}F_x,\delta x. ]

This expression is the linearized dependence of (y) on (x). It also identifies singular points, since the failure of the required inverse can correspond to folds, bifurcations, or changes in the local dimension of the solution set.

In nonlinear optimization, first-order linearization replaces an objective or constraint by its tangent approximation. The derivative identifies locally stationary directions, while second-order information supplied by the Hessian matrix describes curvature omitted by the linear model. Algorithms based on sequential linear or quadratic subproblems repeatedly update the reference state, so their global behavior depends on how those local models are related across successive neighborhoods.

Exactness and limitations

Linearization is exact for affine functions and first-order exact for differentiable functions. It can fail to preserve global properties because distinct nonlinear functions may share the same value and derivative at a reference point. Their linearizations are then identical even though their behavior differs outside an arbitrarily small neighborhood.

At nonhyperbolic equilibria, the linear term may vanish or may leave central directions unresolved. The scalar systems

[ \dot{x}=x^2 \qquad\text{and}\qquad \dot{x}=-x^3 ]

both have zero linearization at the origin, yet their local dynamics are different. Higher-order expansion, center manifold theory, or normal form theory supplies information that first-order analysis omits.

Nondifferentiable functions do not possess a classical linearization at points of nondifferentiability. Generalized derivatives, including the subderivative used in convex analysis, replace the unique tangent map with a set or inequality describing possible first-order behavior. Such constructions extend local approximation beyond smooth settings but are not linearizations in the classical Fréchet-differentiable sense.