Wielandt's Theorem

Let the $n\times n$ Matrix ${\hbox{\sf A}}$ satisfy the conditions of the Perron-Frobenius Theorem and the $n\times n$ Matrix ${\hbox{\sf C}}=c_{ij}$ satisfy

$$\vert c_{ij}\vert\leq a_{ij}$$

for $i,j=1$, 2, ..., $n$. Then any Eigenvalue $\lambda_0$ of ${\hbox{\sf C}}$ satisfies the inequality $\vert\lambda_0\vert\leq R$ with the equality sign holding only when there exists an $n\times n$ Matrix ${\hbox{\sf D}}=\delta_{ij}$ (where $\delta_{ij}$ is the Kronecker Delta) and

$${\hbox{\sf C}}={\lambda_0\over R}{\hbox{\sf D}}{\hbox{\sf A}}{\hbox{\sf D}}^{-1}.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1121, 1979.