Poincaré separation theorem

In mathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem, gives some upper and lower bounds of eigenvalues of a real symmetric matrix B^TAB 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^TB = I_r. Denote by ${\displaystyle \lambda _{i}}$, i = 1, 2, ..., n and ${\displaystyle \mu _{i}}$, i = 1, 2, ..., r the eigenvalues of A and B^TAB, respectively (in descending order). We have

${\displaystyle \lambda _{i}\geq \mu _{i}\geq \lambda _{n-r+i},}$