Poincaré separation theorem explained

In mathematics, 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 B an n × r semi-orthogonal matrix such that B'B = I_r. Denote by

λ _{i} , i = 1, 2, ..., n and \mu _{i} , i = 1, 2, ..., r the eigenvalues of A and B'AB, respectively (in descending order). We haveλ _{i} \geq \mu _{i} \geq λ _{n-r+i} ,

Proof

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 projection of A onto the linear subspace spanned by B, so the above results follow immediately.^[3]

Notes and References

1. [Min-max_theorem#Cauchy_interlacing_theorem]
2. Book: Matrix Differential Calculus with Applications in Statistics and Econometrics. Jan R.. Magnus. Heinz. Neudecker. John Wiley & Sons. 1988. 0-471-91516-5. 209..
3. Book: Richard Bellman. Introduction to Matrix Analysis: Second Edition. 1 December 1997. SIAM. 978-0-89871-399-2. 118–.