Sturmian Separation Theorem

Let ${\hbox{\sf A}}_r=a_{ij}$ be a Sequence of $N$ Symmetric Matrices of increasing order with $i,j=1$, 2, ..., $r$ and $r=1$, 2, ..., $N$. Let $\lambda_k({\hbox{\sf A}}_r)$ be the $k$th Eigenvalue of ${\hbox{\sf A}}_r$ for $k=1$, 2, ..., $r$, where the ordering is given by

$$\lambda_1({\hbox{\sf A}}_r)\geq \lambda_2({\hbox{\sf A}}_r)\geq \ldots \geq \lambda_r({\hbox{\sf A}}_r).$$

Then it follows that

$$\lambda_{k+1}({\hbox{\sf A}}_{i+1})\leq \lambda_k({\hbox{\sf A}}_i)\leq \lambda_k({\hbox{\sf A}}_{i+1}).$$

References

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