Poincaré Separation Theorem

Let $\{{\bf y}^k\}$ be a set of orthonormal vectors with $k = 1$, 2, ..., $K$, such that the Inner Product $({\bf y}^k, {\bf y}^k)=1$. Then set

$${\bf x}=\sum_{k=1}^K u_k{\bf y}^k$$ (1)

so that for any Square Matrix ${\hbox{\sf A}}$ for which the product ${\hbox{\sf A}}{\bf x}$ is defined, the corresponding Quadratic Form is

$$({\bf x},{\hbox{\sf A}}{\bf x})=\sum_{k,l=1}^K u_ku_l({\bf y}^k, {\hbox{\sf A}}{\bf y}^l).$$ (2)

Then if

$${\bf B}_k=({\bf y}^k, {\hbox{\sf A}}{\bf y}^l)$$ (3)

for $k,l=1$, 2, ..., $K$, it follows that

$$\lambda_i({\bf B}_K)\leq \lambda_1({\hbox{\sf A}})$$ (4)

$$\lambda_{K-j}({\bf B}_K)\geq \lambda_{N-j}({\hbox{\sf A}})$$ (5)

for $i=1$, 2, ..., $K$ and $j=0$, 1, ..., $K-1$.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1120, 1979.