mathematics//linear algebra//spectral radius
The spectral radius of a square matrix is the largest magnitude among its eigenvalues, and it is the one number that decides whether repeating a linear step makes things die out or blow up: a discrete-time system, an iterative solver, a training loop or a chain of failures spreading through a network. Writing \(\rho(A)=\max_i|\lambda_i|\), the repeated step \(x_{k+1}=Ax_k\) shrinks every starting vector to zero exactly when \(\rho(A)<1\), and some starting vector grows without bound when \(\rho(A)>1\). The slowest of the decaying modes sets the pace: the distance to zero is multiplied by roughly \(\rho\) per step, so \(\rho=0.98\) per minute means a time constant of about fifty minutes.
The spectral radius of a square matrix is the largest magnitude among its eigenvalues, and it is the one number that decides whether repeating a linear step makes things die out or blow up: a discrete-time system, an iterative solver, a training loop or a chain of failures spreading through a network. Writing ρ(A)=maxi∣λi∣\rho(A)=\max_i|\lambda_i|ρ(A)=maxi∣λi∣, the repeated step xk+1=Axkx_{k+1}=Ax_kxk+1=Axk shrinks every starting vector to zero exactly when ρ(A)<1\rho(A)<1ρ(A)<1, and some starting vector grows without bound when ρ(A)>1\rho(A)>1ρ(A)>1. The slowest of the decaying modes sets the pace: the distance to zero is multiplied by roughly ρ\rhoρ per step, so ρ=0.98\rho=0.98ρ=0.98 per minute means a time constant of about fifty minutes.
The same test appears under many disguises, which is why it pays to recognise it. The stability of a sampled control loop is ρ(A−BK)<1\rho(A-BK)<1ρ(A−BK)<1 (discrete-time stability). Gradient descent near a minimum iterates the matrix I−ηHI-\eta HI−ηH, and its step size limit η<2/λmax\eta<2/\lambda_{\max}η<2/λmax is the condition that this matrix has spectral radius below one. A Markov chain has spectral radius exactly 1, the eigenvalue of its stationary distribution, and its second largest magnitude says how fast it forgets where it started.
A cascade is an unstable eigenvalue on a network.
Linearize how load shed by a failed component is redistributed to its neighbours: if the resulting matrix has spectral radius above one, each failure causes on average more than one new failure, as in a supercritical branching process, and a local trip becomes a cascading failure across a grid or a fleet.
The spectral radius rules the long run and says nothing about the path. For a non-symmetric matrix the norm of AkA^kAk can climb far above 1 before the decay sets in, even with ρ(A)<1\rho(A)<1ρ(A)<1, because the eigenvectors are nearly parallel (non-normal matrix). Formally ∥Ak∥1/k|A^k|^{1/k}∥Ak∥1/k tends to ρ(A)\rho(A)ρ(A) as kkk grows, and only then.
It is bounded above by any matrix norm, which gives cheap safe checks: if every row of a load-redistribution matrix sums to less than one, no eigenvalue can reach one and cascades die out, with no eigenvalue computation at all.
For a continuous system x˙=Ax\dot x=Axx˙=Ax the relevant quantity is the largest real part of the eigenvalues, the spectral abscissa; the spectral radius belongs to the discrete step, which is why the two must not be mixed when a model is discretized.