mathematics//dynamical systems//modes
Modes are the independent patterns into which a coupled linear system splits, each with its own clock, and engineers use them to study such a system one pattern at a time: the vibration of a bridge or a drone frame analysed one frequency at a time, a network of tanks whose disagreements fade at different speeds, a controller that damps the one slow motion that matters. A nonzero **eigenvector** \(v\) of a matrix \(T\) satisfies \(Tv=\lambda v\): the transformation keeps the direction of that pattern and rescales it by the eigenvalue \(\lambda\). What the scale means depends on the matrix: a rate in \(e^{\lambda t}\) for the dynamics \(F\) of \(\dot x=Fx\), a factor per step for a discrete update or a Markov chain, a variance for a covariance matrix, which describes data and governs no time evolution.
Modes are the independent patterns into which a coupled linear system splits, each with its own clock, and engineers use them to study such a system one pattern at a time: the vibration of a bridge or a drone frame analysed one frequency at a time, a network of tanks whose disagreements fade at different speeds, a controller that damps the one slow motion that matters. A nonzero eigenvector vvv of a matrix TTT satisfies Tv=λvTv=\lambda vTv=λv: the transformation keeps the direction of that pattern and rescales it by the eigenvalue λ\lambdaλ. What the scale means depends on the matrix: a rate in eλte^{\lambda t}eλt for the dynamics FFF of x˙=Fx\dot x=Fxx˙=Fx, a factor per step for a discrete update or a Markov chain, a variance for a covariance matrix, which describes data and governs no time evolution.
A mode evolves without changing shape. If x(0)=cvx(0)=cvx(0)=cv, then x(t)=ceλtvx(t)=ce^{\lambda t}vx(t)=ceλtv: every component keeps its proportion and only the amplitude changes. With enough independent eigenvectors, the modal decomposition splits any state into modes, each with its own fate:
x(0)=∑icivi,x(t)=∑icieλitvi,xk=∑iciμikvi.x(0)=\sum_i c_i v_i,\qquad x(t)=\sum_i c_i e^{\lambda_i t}v_i,\qquad x_k=\sum_i c_i\mu_i^k v_i .x(0)=i∑civi,x(t)=i∑cieλitvi,xk=i∑ciμikvi.
In continuous time a mode decays when Reλi<0\operatorname{Re}\lambda_i<0Reλi<0; in discrete time when ∣μi∣<1|\mu_i|<1∣μi∣<1. A complex pair μ=re±iθ\mu=re^{\pm i\theta}μ=re±iθ is an oscillation: the modulus is how much survives each step, the angle how far it turns, that is, its frequency.
Two zones of a workshop, thermally coupled and updated each minute by [0.950.03\0.030.95]\begin{bmatrix}0.95&0.03\0.03&0.95\end{bmatrix}[0.950.03\0.030.95], show it plainly. The eigenvectors are (1,1)(1,1)(1,1) and (1,−1)(1,-1)(1,−1), with eigenvalues 0.98 and 0.92: the mean temperature of the two zones relaxes slowly, with a time constant of about 50 minutes, and the difference between them vanishes in about 12. These are the common mode and the differential mode, each with its own clock; a chain of three tanks has three such patterns (three-tank network).
Eigenvectors decouple a linear system into independent modes, and each eigenvalue says whether its mode dies or explodes.
The dominant mode, the slowest one (closest to the imaginary axis, or to the unit circle), governs the long run; what the eigenvalues do not tell is how large the transient gets first.
The Jacobian provides the matrix whose modes are read here, and the scalar decay is the one-mode case; in a structure the same decomposition gives mode shapes and natural frequencies (modal analysis). (A saddle point of a loss landscape and a saddle equilibrium are the same eigen-structure read on different matrices.)
?Does every matrix split cleanly into modes?
Not always. Real symmetric matrices admit an orthonormal basis of real eigenvectors, and then ci=viTx(0)c_i=v_i^{\mathsf T}x(0)ci=viTx(0). A non-symmetric matrix can have complex eigenvalues, non-orthogonal eigenvectors or fail to be diagonalizable; in a Jordan block factors like teλtte^{\lambda t}teλt appear. With nearly parallel eigenvectors, typical of chains of subsystems in series, the matrix can amplify a transient many times before every mode decays (non-normal matrix): the spectrum describes the asymptotics, the singular values of AkA^kAk the transient.