Poincaré Separation Theorem

The Poincaré separation theorem bounds the eigenvalues of a projected symmetric matrix between eigenvalues of the original matrix. 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, ..., .