“Poincaré separation theorem”的意思、由来-开放百科全书

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Poincaré separation theorem

释义

Proof
References

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

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

1. ^{{cite book |title=Matrix Differential Calculus with Applications in Statistics and Econometrics|first1=Jan R.|last1=Magnus|first2=Heinz|last2=Neudecker|publisher=John Wiley & Sons|year=1988|isbn=0-471-91516-5|pages=209.}}
2. ^{{cite book|author=Richard Bellman|title=Introduction to Matrix Analysis: Second Edition|url=https://books.google.com/books?id=sP8J4oqwlLkC&pg=PA118|date=1 December 1997|publisher=SIAM|isbn=978-0-89871-399-2|pages=118–}}

2 : Inequalities|Matrix theory

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