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 = I_r. Denote by $\lambda _{i}$ , i = 1, 2, ..., n and $\mu _{i}$ , i = 1, 2, ..., r the eigenvalues of A and B'AB, respectively (in descending order). We have

\lambda _{i}\geq \mu _{i}\geq \lambda _{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.[1] 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.[2]

References

Magnus, Jan R.; Neudecker, Heinz (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons. p. 209. ISBN 0-471-91516-5.
Richard Bellman (1 December 1997). Introduction to Matrix Analysis: Second Edition. SIAM. pp. 118–. ISBN 978-0-89871-399-2.

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[1] Magnus, Jan R.; Neudecker, Heinz (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons. p. 209. ISBN 0-471-91516-5.

[Bellman1997-2] Richard Bellman (1 December 1997). Introduction to Matrix Analysis: Second Edition. SIAM. pp. 118–. ISBN 978-0-89871-399-2.