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Consider the space of uniformly distributed $2 \times 2$ real matrices \begin{equation} \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right), \end{equation} with the restriction that the larger of the matrices' two singular values (that is, operator [Schatten-$\infty$] norm) is bounded above by 1. Now, the (apparently very challenging) problem is to construct the function $\tilde{\chi}_1(\epsilon)= \tilde{\chi}_1(1/\epsilon)$, for $\epsilon >0$, giving the probability that the companion matrix \begin{equation} \left( \begin{array}{cc} a & b \epsilon \\ \frac{c}{\epsilon } & d \\ \end{array} \right) \end{equation} also has its larger singular value bounded above by 1. This problem has very recently been solved (I can give the reference), but I would hope to encourage a (fresh) Mathematica formulation/solution. This might be helpful in trying to extend the problem to the (presently unsolved) one of constructing the function $\tilde{\chi}_2(\epsilon)$ corresponding to the $2 \times 2$ matrices with complex entries.

This problem can be easily formulated in a few lines of Mathematica code (using the SingularValueList and Integrate--with a Boole [4-dimensional] condition--commands, for instance). However, it appears to be--at least, with my somewhat limited resources--too computationally demanding to arrive at a solution. Perhaps there are other set-ups/transformations/workarounds to investigate. In any case, I thought it might be some "fun" to try for a few Mathematica devotees--and an interesting test for the power/limitations of Mathematica. Further, let me point out the reference in which the solution was found (eq. (9) in https://arxiv.org/pdf/1610.01410.pdf, also the succinct Conclusion section [p. 17]). It is not clear if these authors employed Mathematica, and, if so, in what manner.

I added this last paragraph in response to the question having been put on hold as requiring "either advice from Wolfram support or the services of a professional consultant".

Here (per the comment of Jim Baldwin) is some Mathematica coding for the problem:

H1 = {{a, b}, {c, d}};
H2 = {{a, e b}, {c/e, d}};
v1 = Expand[PowerExpand[FullSimplify[(SingularValueList[H1] /. Conjugate[f_] -> f)[[2]]]^2]]
v2 = Expand[PowerExpand[FullSimplify[(SingularValueList[H2] /. Conjugate[f_] -> f)[[2]]]^2]]
s1 = Integrate[Boole[v1 < 1], {a, -1, 1}, {b, -1, 1}, {c, -1, 1}, {d, -1, 1}]
s2 = Integrate[Boole[v1 < 1 && v2 < 1 && e > 0], {a, -1, 1}, {b, -1, 1}, {c, -1, 1}, {d, -1, 1}]

The computation of the variable "s2" should give the desired function $\tilde{\chi}_1(\epsilon)$, but it seems too demanding a task to complete. ("H1" and "H2" are the indicated matrices, while "v1" and "v2" are the squares of the corresponding larger singular values. These serve as constraints in the four-dimensional integrations.) Perhaps the CylindricalDecomposition command might be helpful, but some preliminary efforts along such lines have not been fruitful.

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    $\begingroup$ I don't know the importance of this question but it sounds like you're asking others to solve a research problem of yours or searching for a collaborator. I don't think this site is built for such things. $\endgroup$ – JimB Jan 20 '17 at 5:22
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    $\begingroup$ "This problem can be easily formulated in a few lines of Mathematica code (using the SingularValueList and Integrate--with a Boole [4-dimensional] condition--commands, for instance). " Showing your code would generate much more interest. $\endgroup$ – JimB Jan 20 '17 at 17:59