In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest …

Derivation from Perron-Frobenius Theory suppose A is Metzler, and choose τ (e.g., τ = 1 − mini Aii) s.t. τ I + A ≥ 0 by PF theory, τ I + A has PF eigenvalue λpf , with associated right and left eigenvectors v ≥ …

The Perron-Frobenius theorem tells us that if we increase any matrix element in a primitive matrix, A, then the dominant eigenvalue r increases. But by how much?

1. Introduction We begin by stating the Frobenius-Perron Theorem: .1 (Frobenius-Perron). Let B be an n × n matrix with no negative real entries. Then

Since the r-eigenspace is one-dimensional, x = cv. Thus (∀i) xi = cvi. This is possible if and only if c > 0. Conclude that x = cv is another positive r-eigenvector if and only if c > 0. This completes the proof of …

The theorem generalizes to situations considered in chaos theory, where products of random matrices are considered which all have the same distribution but which do not need to be independent.

Dec 22, 2025 · Weisstein, Eric W. "Perron-Frobenius Theorem." From MathWorld --A Wolfram Resource. https://mathworld.wolfram.com/Perron-FrobeniusTheorem.html.

What is the Perron-Frobenius theorem? The Perron-Frobenius theorem places constraints on the largest eigen-values and positive eigenvectors of matrices with non-negative entries.

Jun 16, 2025 · The Perron-Frobenius theorem # For a square nonnegative matrix A, the behavior of A k as k → ∞ is controlled by the eigenvalue with the largest absolute value, often called the dominant …

Theorem 2. If all su ciently high powers of A are real and positive, then the extremal eigenvalues of A are real and positive (i.e., if is extremal, then = (A) > 0).