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Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.

For each $\alpha$, it is given that $L(\alpha)$ is a symmetric matrix, and so it has a basis of eigenvectors, and $n$ eigenvalues (not necessarily distinct). We define a function $f(\alpha)$ to be the smallest eigenvalue of $L(\alpha)$.

It is also given that the elements of $L(\alpha)$ are smooth functions of $\alpha$ for each $\alpha$. Therefore, as the eigenvalues are the roots of a polynomial whose coefficients are also smooth functions of $\alpha$, $f(\alpha)$ is a smooth function of $\alpha$ for all but a few values of $\alpha$ (this follows from the implicit function theorem).

The problem I am having is to try to find the global minimum of $f(\alpha)$. Usually what one would do is to take its derivative with respect to alpha, and set that equal to 0, but that is actually not possible in the present case because there is no way to get an explicit formula for $f(\alpha)$.

A possible idea that I had was to get some kind of sequence which converges to the smallest eigenvalue of $L(\alpha)$, and hope that this sequence converges uniformly in $\alpha$, but I have not yet found such a thing.

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I don't find a question here. –  Denis Serre Feb 16 '11 at 9:27
    
I belive he wants a method to minimalize $f(\alpha)$. Is it clear that this minimum exists? –  András Bátkai Feb 16 '11 at 9:32
    
over a compact set of $\alpha$'s. Otherwise certainly not: $\alpha \mapsto L (\alpha) = \alpha$ viewed as a $1\times1$ matrix does not have a minimum... –  Stefan Waldmann Feb 16 '11 at 9:51
    
Educated guess: The search is for the global minimum of the absolute value of the smallest eigenvalue. –  Tim van Beek Feb 16 '11 at 10:13
    
I did not say this in the above, but it is implicit that I am assuming the existence of a global minimum. Just like you can find the global minimum of a function f: R -> R in principle by solving f'=0 should it exist, is there a practical way to in principle find the global minimum of this function $f(\alpha)$ given that it exists? –  Eric Haengel Feb 16 '11 at 12:44

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