Perron-Frobenius theorem

When a matrix has all entries positive, one eigenvalue stands out: it is real, positive, strictly larger in magnitude than every other eigenvalue, and its eigenvector can be chosen with all-positive entries. This dominant pair is unique. That is the Perron-Frobenius theorem, and it is the reason iterating such a matrix steers any positive start toward one preferred direction.

Precisely, if A > 0 (entrywise) then there is a simple eigenvalue r > 0, the Perron root, equal to the spectral radius, with a strictly positive eigenvector v (the Perron vector). Frobenius extended this to nonnegative matrices that are irreducible (the associated graph is strongly connected): r is still positive with a positive eigenvector, though for periodic matrices other eigenvalues may also sit on the circle of radius r.

The positivity of the eigenvector is the magical part. For most matrices an eigenvector mixes positive and negative signs, but a Perron vector is all one sign, so it can be read as a distribution or a ranking, never canceling itself out.

Why it matters: this single theorem underwrites PageRank, the steady state of irreducible Markov chains, the leading mode of population models (Leslie matrices), and economic input-output equilibria. The caveat is irreducibility: if the graph splits into pieces that cannot reach each other, the clean unique-positive-eigenvector guarantee breaks.