Poincaré Separation Theorem

Let be a set of orthonormal vectors with , 2, ..., , such that the Inner Product . Then set

 (1)

so that for any Square Matrix for which the product is defined, the corresponding Quadratic Form is
 (2)

Then if
 (3)

for , 2, ..., , it follows that
 (4)

 (5)

for , 2, ..., and , 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.