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Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing? In particular my matrix is $$-\frac{ik}{4\pi}I+diag[a_1,\ldots,a_n]+A(k)$$ where $I$ is the identity matrix, $\alpha_i$ are real constants and $$\[A(k)\]_{jj}=0 $$ $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}}{4\pi|y_j-y_l|}\, j\neq l $$ with $y_i$ point in $\mathbb{R}^3$.

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You need to explain what you mean by a derivative of a matrix, since as defined they are not functions. –  Chris Godsil Mar 6 '13 at 11:53
    
I have a matrix wich depends by a parameter $k$; so I have to derive the entries to respect to $k$. –  Mario Mar 6 '13 at 11:56
    
If I understand well, your matrix is really explicit and of size 3. Have you tried computations, either symbolic with a CAS, or numerical ? That may give you a good idea of what is happening. (Maybe plot the eigenvalues against k, for some particular points y ?) –  Thomas Richard Mar 6 '13 at 12:48
    
My matrix is square of size $n$; I don't have tried any calculation because in the book where I'm studying the author invokes the calculation of the derivative of the matrix, but I don't really understand his reasoning. –  Mario Mar 6 '13 at 13:09
    
What's $i$? Is this $\sqrt{-1}$? Then it what sense your derivative is positive-definite? –  Alexandre Eremenko Mar 6 '13 at 13:35
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2 Answers

Look at: Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416, (pdf).

There is an overview on available results.

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Sorry for posting this as an answer, I cannot leave comment.

Are you sure your matrix is $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}}{4\pi|y_j-y_l|}\, j\neq l$$ and not $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}-1}{4\pi|y_j-y_l|}\, j\neq l$$

If you want to extend some properties of the derivative of a matrix to its eigenvalue, the eigenvectors have to be independent of the derivation variable.

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I'm sure that the matrix I've written is correct. I want to know if it is invertible and so if it has no zero eigenvalue; it should be that if $k<0$ the matrix can have a zero eigenvalue. –  Mario Mar 6 '13 at 15:17
    
Do you have additional information on the $\alpha_i$ or they are of general form? –  Hicham Mar 6 '13 at 15:29
    
They are general –  Mario Mar 6 '13 at 15:36
    
I don't see how to make further progress, could you give the reference of the book you've mentioned? –  Hicham Mar 6 '13 at 16:08
    
"Solvable models in quantum mechanics" by S.Albeverio –  Mario Mar 6 '13 at 16:09
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