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I have the following list $\beta$ with 5 equations

{
 (8 + n) λ[1]^2 + (4 (-1 + n) (1 + n)^4 λ[2]^2)/n^6 + λ[1] (-1 + (4 (-1 + n) (1 + n)^2 λ[2])/n^3) +m λ[5]^2,
 (λ[2] (-21 λ[2] - 33 n λ[2] - 3 n^2 λ[2] +n^3 (-1 + 12 λ[1] + 9 λ[2])))/n^3,
 (8 + m) λ[3]^2 + (4 (-1 + m) (1 + m)^4 λ[4]^2)/m^6 + λ[3] (-1 + (4 (-1 + m) (1 + m)^2 λ[4])/m^3) +n λ[5]^2,
 (λ[4] (-21 λ[4] - 33 m λ[4] - 3 m^2 λ[4] +m^3 (-1 + 12 λ[3] + 9 λ[4])))/m^3,
 λ[5] (-1 + (2 + n) λ[1] + (2 (-1 + n) (1 + n)^2 λ[2])/n^3 + 2 λ[3] + m λ[3] + (2 (-1 + m) (1 + m)^2 λ[4])/m^3 + 4 λ[5])
}

where m, n are integers as input.

Every time I assign some integer values to $m,n$ I can solve $\beta=0$ to get λ[1], λ[2], λ[3], λ[4], λ[5].

With these ingredients, I can evaluate the eigenvalues of the matrix

Table[D[\beta[[i]], λ[j]], {i, 5}, {j, 5}] `

to check whether all eigenvalues are positive.

In summary, I want to know whether there exists some positive integers $m,n$ such that the eigenvalues of the matrix mentioned above are all positive.

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    $\begingroup$ hard to suggest specific solution since you did not show the function itself. But try FindMaximum or MaxValue or NMaxValue there is also many questions here that ask to find max of function. You can search these. $\endgroup$
    – Nasser
    Jan 9 at 21:16
  • $\begingroup$ @Nasser Thank you for your comment. I tried to let this question be more specific. It would be very helpful if you can make further comment : ) $\endgroup$
    – Vayne
    Jan 9 at 21:45
  • $\begingroup$ @Vayne A couple of things: 1) setting your expression to zero gives you three equations and five unknowns (lambda 1 through 5); that's not enough to solve for them explicitly, isn't it? 2) The $\beta_{i}$ you showed does not depend on $i$, but on $n$. Should we understand $n$ to be $i$? 3) Please post expressions as MMA code, not $\LaTeX$, so we don't have to type them in ourselves. 4) Please come up with a better title for your question that describes the problem you are having. Summarizing one's problem in a few words sometimes helps clarify it for you as well, so it may be useful. $\endgroup$
    – MarcoB
    Jan 9 at 22:12
  • $\begingroup$ @MarcoB, thanks for the suggestions. I have made some modifications to the question. I will try to make more edits later when I can figure out a better way . $\endgroup$
    – Vayne
    Jan 10 at 17:02
  • 1
    $\begingroup$ This might be intractable. $\endgroup$ Jan 10 at 20:28

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