---- > [!theorem] Theorem. ([[Perron-Frobenius Theorem for primitive matrices]]) > Let $A$ be a [[primitive matrix]] over $\mathbb{C}$. Then > 1. The [[spectral radius]] $r:=\rho(A)$, called the **Perron root**, is in fact an [[eigenvalue]] of $A$. Moreover, it is *strictly positive*. > - i.e., the largest magnitude [[eigenvalue]] is real and positive. > 2. $r$ is [[simple eigenvalue|simple]] > 3. *Uniqueness of magnitude*: For any other [[eigenvalue]] $\lambda$, $|\lambda|<r=\rho(A)$ [^1]: > 4. $A$ has a unit-norm (right) [[positive matrix|positive]] [[eigenvector]] $v$, called a **right Perron vector**, corresponding to $r$: $Av = rv$. > 5. $A$ has a unit-norm [[positive matrix|positive]] [[eigenvector]] $w$, called a **left Perron vector**, for which $w^{\top}A=rw^{\top}$. > 6. $v$ and $w$ are unique, in the sense that all other right resp. left unit-norm [[eigenvector|eigenvectors]] of $A$ have negative and/or non-real elements. > > **Corollary.** Because a [[primitive matrix]] $A$ has a dominating [[eigenvalue]], the [[power iteration]] converges to the **Perron vector** (with no sign issues if initialized with a positive [[vector]]). > ![[CleanShot 2023-11-20 at 19.01.38.jpg]] [^1]: This is the key strengthening over the [[Perron-Frobenius Theorem for irreducible matrices]], which itself is a strengthening over the [[Perron-Frobenius Theorem for square nonnegative matrices]]. > [!proof]- Proof. ([[Perron-Frobenius Theorem for primitive 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 > ```