Poincaré separation theorem
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In mathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem, gives some upper and lower bounds of
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of a real
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
''B'AB'' that can be considered as the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
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 In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then ...
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_,


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''. From the geometric point of view, ''B'AB'' can be considered as the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of ''A'' onto the linear subspace spanned by ''B'', so the above results follow immediately.


References

{{DEFAULTSORT:Poincare Separation Theorem Inequalities Matrix theory