Generalized Eigenvalue Problem

[JohnSimpson]

Consider a generalized Eigenvalue problem Av = \lambda Bv
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

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

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

?

 

[AlephZero]

For any vector $x$, the Rayleigh quotient $x^T A x / x^T B x \ge x^T A x / x^T B_{max} x$.

And $\lambda_{min} = \min_x x^T A x / x^T B x$.

Physically it is "obvious" if A is a stiffness matrix and B is a mass matrix. Increasing the mass (by making all the diagonal entries of B equal to the biggest) must reduce the vibration freqencies.

Another way to attack this would be to treat it as a perturbation of the original problem, i.e. let $B_{max} = B + D$. IIRC there is some nice theory about this, but I'm not energetic enough to start looking it up right now.