Consider a generalized Eigenvalue problem Av = \lambda Bv(adsbygoogle = window.adsbygoogle || []).push({});

where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries.

It is clear that the generalized eigenvalues will be nonnegative. What else can one say about the eigenvalues of the generalized problem in terms of the eigenvalues of $A$ and the diagonals of $B$? Equivalently, what else can one say about the eigenvalues of B^{-1}A?

It seems reasonable (skipping over zero eigenvalues) that

[tex]

\lambda_{min}(B^{-1}A) \geq \lambda_{min}(A)/B_{max}

[/tex]

but I am unable to see how one could rigorously show this, and it is perhaps a conservative bound. Equivalently again, what could one say about the eigenvalues of

[tex]

B^{-1/2}AB^{-1/2}

[/tex]

?

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# Generalized Eigenvalue Problem

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