mathematics//dynamical systems//modes

Which patterns does a matrix preserve, and what does the factor it applies to them mean? 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 \(\lambda\). The meaning of the scale depends on what \(T\) represents.

Which patterns does a matrix preserve, and what does the factor it applies to them mean? 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. The meaning of the scale depends on what TTT represents.

Same equation, different questions. FFF in x˙=Fx\dot x=Fxx˙=Fx: the eigenvector is a dynamic pattern of the state; the eigenvalue a time rate in eλte^{\lambda t}eλt.

LLL in x˙=−Lx\dot x=-Lxx˙=−Lx: a pattern of disagreement over a network; a decay rate in e−λte^{-\lambda t}e−λt when LLL has units of rate.

PPP in a Markov chain: a mode of the transition operator; a per-step factor μ\muμ, where side and convention matter.

Σ\SigmaΣ of data (covariance matrix): a direction in variable space; the projected variance for a unit eigenvector. Eigenvectors are not always equilibria, nor always vibrations. A covariance does not by itself govern the time evolution of a system.

Why a mode evolves without changing shape. If Fv=λvFv=\lambda vFv=λv and x(0)=cvx(0)=cvx(0)=cv, then x(t)=ceλtvx(t)=ce^{\lambda t}vx(t)=ceλtv: every component keeps its proportion within the pattern and only the amplitude changes. With enough independent eigenvectors any state decomposes as

x(0)=∑icivi,x(t)=∑icieλitvi.x(0)=\sum_i c_i v_i,\qquad x(t)=\sum_i c_i e^{\lambda_i t}v_i.x(0)=i∑​ci​vi​,x(t)=i∑​ci​eλi​tvi​.

With an orthonormal basis, ci=viTx(0)c_i=v_i^{\mathsf T}x(0)ci​=viT​x(0). In a general basis the inverse of the eigenvector matrix is used, not simply its transpose.

Important limits. Real symmetric matrices admit an orthonormal basis of real eigenvectors. A non-symmetric matrix can have complex eigenvalues, non-orthogonal eigenvectors or fail to be diagonalizable, without necessarily showing all those problems. In a Jordan block factors like teλtte^{\lambda t}teλt appear. Non-normal matrices can amplify transiently even when every eigenvalue has negative real part: the spectrum describes the asymptotics, not the whole transient. A complex pair α±iω\alpha\pm i\omegaα±iω of a real matrix is a modal plane with growth or decay and rotation; the real pattern is rebuilt from the conjugate pair.

Network example. For a chain of three tanks, (1,1,1)T(1,1,1)^{\mathsf T}(1,1,1)T is the constant mode; (1,0,−1)T(1,0,-1)^{\mathsf T}(1,0,−1)T and (1,−2,1)T(1,-2,1)^{\mathsf T}(1,−2,1)T describe different disagreements. Under diffusion their amplitudes vanish at different speeds (three-tank network).

The Jacobian provides the matrix whose modes are read here; the scalar decay is the one-mode case.