Inmathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem,[1] gives some 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 Bann × r semi-orthogonal matrix such that B'B = Ir. Denote by 
An algebraic proof, based on the variational interpretation of eigenvalues, has been published in Magnus' Matrix Differential Calculus with Applications in Statistics and Econometrics.[2] From the geometric point of view, B'AB can be considered as the orthogonal projectionofA onto the linear subspace spanned by B, so the above results follow immediately.[3]
