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Poincaré separation theorem

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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 = Ir. Denote by , i = 1, 2, ..., n and , i = 1, 2, ..., r the eigenvalues of A and B'AB, respectively (in descending order). We have

Proof

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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]

References

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  1. ^ Min-max_theorem#Cauchy_interlacing_theorem
  2. ^ 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.
  3. ^ Richard Bellman (1 December 1997). Introduction to Matrix Analysis: Second Edition. SIAM. pp. 118–. ISBN 978-0-89871-399-2.