---- > [!theorem] Theorem. ([[Perron-Frobenius Theorem for irreducible matrices]]) > Let $A$ be a [[nonnegative matrix|nonnegative]] [[irreducible matrix]]. Then > 1. The [[spectral radius]] $r:=\rho(A)$, called the **Perron root**, is in fact an [[eigenvalue]] of $A$. Moreover, it is *strictly positive* [^1]. > - i.e., the largest magnitude [[eigenvalue]] is real and positive. > 2. $r$ is [[simple eigenvalue|simple]] [^2] > 1. $A$ has a unit-norm (right) [[positive matrix|positive]] [[eigenvector]] $v$, called a **right Perron vector**, corresponding to $r$: $Av = rv$. > 2. $A$ has a unit-norm [[positive matrix|positive]] [[eigenvector]] $w$, called a **left Perron vector**, for which $w^{\top}A=rw^{\top}$. > 3. $v$ and $w$ are unique, in the sense that all other unit-norm left resp. right [[eigenvector|eigenvectors]] have negative and/or non-real elements. [^3] [^1]: This is the first key strengthening from the [[Perron-Frobenius Theorem for square nonnegative matrices]]. [^2]: This is the second key strengthening from the [[Perron-Frobenius Theorem for square nonnegative matrices]]. [^3]: This is the third key strengthening, and it is crucial to notions such as [[eigenvector centrality]]. ![[CleanShot 2023-11-20 at 18.43.54.jpg]] > [!proof]- Proof. ([[Perron-Frobenius Theorem for irreducible matrices]]) > ~ ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```