Karpelevič Region Viewer

NIEP

Karpelevič Region Viewer Nonnegative Inverse Eigenvalue Problem

Matrix Order (n) Stochastic n × n

2 4 6 8 12 16

Complex Number (z) Click / Drag on Plane

Real Part Re(z)

Imaginary Part Im(z)

Modulus |z|

Argument Arg(z) / π

Inside K_n |z| = 0.632

Power Up Dynamics λ ∈ K_n ⟹ λᵏ ∈ K_n

Current Power: z¹

k=1 k=5 k=10 k=15 k=20 k=25

Power Orbit Trajectory Powers Table

Power	Value (Re + Im i)	\|zᵏ\|	Arg	Status

Mathematical Theory ▼

The Karpelevič Region (𝒦_n) is the set of all eigenvalues of n × n stochastic matrices (or nonnegative matrices with Perron eigenvalue 1).

Boundary Structure: The boundary intersects the unit circle {|z|=1} at roots of unity e^2πi·p/q where p/q are Farey fractions of order n (with 0 ≤ p ≤ q ≤ n and gcd(p,q)=1).

Ito Polynomial Arcs: Consecutive Farey neighbors p/q < r/s (with q < s) are connected by curvilinear boundary arcs satisfying Ito's polynomial equation:

tˢ·(tᑫ - (1-α))ᵏ = αᵏ·tᑫᵏ, for α ∈ [0, 1], where k = ⌊n/q⌋.

The Power Property: If A is an n × n stochastic matrix with eigenvalue λ, then Aᵏ is also an n × n stochastic matrix with eigenvalue λᵏ. Consequently, whenever λ ∈ 𝒦_n, all powers λᵏ ∈ 𝒦_n for all integers k ≥ 1.

Re: 0.000, Im: 0.000 • n = 4

𝒦_n Boundary

|z| = 1

Power Orbit (zᵏ)

Farey Roots