---- > [!theorem] Theorem. ([[Perron-Frobenius Theorem for square nonnegative matrices]]) > Let $A$ be a square, [[nonnegative matrix]]. Then > 1. The [[spectral radius]] $r:=\rho(A) \geq 0$, called the **Perron root**, is in fact an [[eigenvalue]] of $A$ > - i.e., the largest magnitude [[eigenvalue]] is real and nonnegative. > 2. $A$ has a (right) [[nonnegative matrix|nonnegative]] [[eigenvector]] $v$, called a **right Perron vector**, corresponding to $r$: $Av = rv$. > 3. $A$ has a left [[nonnegative matrix|nonnegative]] [[eigenvector]] $w$, called a **left Perron vector**, for which $w^{\top}A=rw^{\top}$. ![[CleanShot 2023-11-20 at 18.43.32.jpg]] > [!proof]- Proof. ([[Perron-Frobenius Theorem for square nonnegative 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 > ```