Ostrowski's Theorem

Let ${\hbox{\sf A}}=a_{ij}$ be a Matrix with Positive Coefficients and $\lambda_0$ be the Positive Eigenvalue in the Frobenius Theorem, then the $n-1$ Eigenvalues $\lambda_j\not=\lambda_0$ satisfy the Inequality

$$\vert\lambda_j\vert\leq\lambda_0 {M^2-m^2\over M^2+m^2},$$

where

$$M = \max_{i,j} a_{ij}$$

$$m = \min_{i,j} a_{ij}$$

and $i,j=1$, 2, ..., $n$.

See also Frobenius Theorem

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1121, 1980.