What is Poincaré separation theorem?

In mathematics, the Poincaré separation theorem gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. The theorem is named after Henri Poincaré.

More specifically, let A be an n × n real symmetric matrix and B an n × r semi-orthogonal matrix such that B'B = Ir. Denote by λ i {\displaystyle \lambda _{i}} , i = 1, 2, ..., n and μ i {\displaystyle \mu _{i}} , i = 1, 2, ..., r the eigenvalues of A and B'AB, respectively. We have