While explaining the application of Geometric programming to Minimizing Spectral radius Boyd says that $\lambda_{pf}$ can also be characterized as: $\operatorname{inf}\{\lambda|\exists{v}>0, Av\leq\lambda{v}\}$
According to my understanding Perron Frobenius Eigenvalue is the Eigenvalue with largest absolute value. These two definitions seems conflicting to me.
Can someone explain?
Presumably $A$ is (entrywise) positive or it is both irreducible and nonnegative. By Perron-Frobenius theorem, the spectral radius $\rho(A)$ is an eigenvalue of $A$ and there exists a positive left eigenvector $u$ corresponding to the eigenvalue $\rho(A)$. Now, if $Av\le\lambda v$ for some positive $v$, then $\rho(A)u^Tv=u^TAv\le\lambda u^Tv$, i.e. $\rho(A)\le\lambda$. Therefore $\rho(A)\le\inf\{\lambda\mid\exists v>0,\,Av\le\lambda v\}$. However, as $Av=\rho(A)v$ for the (right) Perron vector of $A$, the inequality is sharp.