Linear dynamical system

Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. In a linear dynamical system, the variation of a state vector (an N {displaystyle N} -dimensional vector denoted x {displaystyle mathbf {x} } ) equals a constant matrix(denoted A {displaystyle mathbf {A} } ) multiplied by x {displaystyle mathbf {x} } . This variation can take two forms: either as a flow, in which x {displaystyle mathbf {x} } varies continuously with time or as a mapping, in which x {displaystyle mathbf {x} } varies in discrete steps These equations are linear in the following sense: if x ( t ) {displaystyle mathbf {x} (t)} and y ( t ) {displaystyle mathbf {y} (t)} are two valid solutions, then so is any linear combination of the two solutions, e.g., z ( t )   = d e f   α x ( t ) + β y ( t ) {displaystyle mathbf {z} (t) {stackrel {mathrm {def} }{=}} alpha mathbf {x} (t)+eta mathbf {y} (t)} where α {displaystyle alpha } and β {displaystyle eta } are any two scalars. The matrix A {displaystyle mathbf {A} } need not be symmetric. Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems. If the initial vector x 0   = d e f   x ( t = 0 ) {displaystyle mathbf {x} _{0} {stackrel {mathrm {def} }{=}} mathbf {x} (t=0)} is aligned with a right eigenvector r k {displaystyle mathbf {r} _{k}} of the matrix A {displaystyle mathbf {A} } , the dynamics are simple where λ k {displaystyle lambda _{k}} is the corresponding eigenvalue;the solution of this equation is