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Let $a,b,c$ and $d$ such that the matrix $ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathbb C^{2,2} $ has to two distinct eigenvalues. Give a neighbourhood $U\ne\mathbb C$ in which are both eigenvalues. How do one one handle this problem? I thought about Gerschgorin circles, but I am not sure. Any kind of help is appreciated!

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  • $\begingroup$ A neighbourhood of what? $\endgroup$
    – Bernard
    Feb 16, 2018 at 12:57
  • $\begingroup$ it is an old exam question. There is nothing else stated $\endgroup$
    – user519338
    Feb 16, 2018 at 12:58
  • $\begingroup$ I guess $U$ should just be an open set containing both eigenvalues. Note that the absolute value of any eigenvalue is smaller than the Frobenius norm. That is a pretty common inequality. $\endgroup$
    – MooS
    Feb 16, 2018 at 12:59
  • $\begingroup$ So probably it's a neighbourhood of each coefficient. Or a neighbourhood of the matrix for some matrix norm, but in this case it's a neighbourhood in $\mathbf C^{2,2}$. $\endgroup$
    – Bernard
    Feb 16, 2018 at 13:00
  • $\begingroup$ After enlarging $U$, $U$ is a neighbourhood of anything you want. Basically the question asks you to bound the eigenvalues in terms of $a,b,c,d$. $\endgroup$
    – MooS
    Feb 16, 2018 at 13:02

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