 ##  [Perron-Frobenius Theorem](/perron-frobenius-theorem) 

  ##  [Perron-Frobenius Theorem](https://mathlogic.quantumdictionary.io/perron-frobenius-theorem-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A collection of results about positive or nonnegative matrices and operators: under positivity/irreducibility hypotheses there exists a unique dominant real eigenvalue (the Perron root) with an associated strictly positive eigenvector and related spectral properties.