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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
\end{displaymath} (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).
\end{displaymath} (2)

Then if
{\bf B}_k=({\bf y}^k, {\hbox{\sf A}}{\bf y}^l)
\end{displaymath} (3)

for $k,l=1$, 2, ..., $K$, it follows that
\lambda_i({\bf B}_K)\leq \lambda_1({\hbox{\sf A}})
\end{displaymath} (4)

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

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


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

© 1996-9 Eric W. Weisstein