Specifically, I want to check the positive semi-finiteness of the following 6X6 symbolic matrix

```
{{-2 (-((11 m1^2 - 24 m1 m3 + 72 m3^2)/(
144 k^2 m1^2 m3^2 (Sigma)^2)) - 2 (Sigma)^2),
I - (m1 - 6 m3)/(6 k m1 m3 (Sigma)^2), (
3/m1 + 1/m3 + (12 (Sigma)^4)/m2)/(6 k (Sigma)^2), 2 (Sigma)^2,
1/(8 k^2 m3^2 (Sigma)^2) + 4 (Sigma)^2, (m1 + 12 m3)/(
12 k m1 m3 (Sigma)^2)}, {-I - (m1 - 6 m3)/(6 k m1 m3 (Sigma)^2),
1/(Sigma)^2, 1/(2 (Sigma)^2), 0, 0, 1/(Sigma)^2}, {(
3/m1 + 1/m3 + (12 (Sigma)^4)/m2)/(6 k (Sigma)^2), 1/(
2 (Sigma)^2), 1/(Sigma)^2 + (4 (Sigma)^2)/(k^2 m2^2),
I + (4 (Sigma)^2)/(k m2), (1/m3 + (16 (Sigma)^4)/m2)/(
4 k (Sigma)^2), 1/(Sigma)^2}, {2 (Sigma)^2,
0, -I + (4 (Sigma)^2)/(k m2), 4 (Sigma)^2, 4 (Sigma)^2,
0}, {1/(8 k^2 m3^2 (Sigma)^2) + 4 (Sigma)^2, 0, (
1/m3 + (16 (Sigma)^4)/m2)/(4 k (Sigma)^2), 4 (Sigma)^2,
1/(8 k^2 m3^2 (Sigma)^2) + 6 (Sigma)^2,
I + 1/(4 k m3 (Sigma)^2)}, {(m1 + 12 m3)/(12 k m1 m3 (Sigma)^2),
1/(Sigma)^2, 1/(Sigma)^2, 0, -I + 1/(4 k m3 (Sigma)^2), 3/(
2 (Sigma)^2)}}
```

What depends on the 5 parameters: $ m_ {1} $, $ m_ {2} $, $ m_ {3} $, k $, $ sigma $, which are all positive, that is to say

$ m_ {1}> 0, m_ {2}> 0, m_ {3}> 0, k> 0, sigma> 0 tag {1} $ .

I try to apply the methodology proposed in (Check if a symbolic matrix is semi-definite positive) but Mathematica continues to perform the calculation and gives no results.

Then, as another way of solving the problem, I tried to numerically inspect the minimum and maximum eigenvalue values of the matrix above, using `NMinimize()`

and `NMaximize()`

subject to the constraints indicated in (1). My code is

```
s=Simplify(Eigenvalues({{-2 (-((11 m1^2-24 m1 m3+72 m3^2)/(144 k^2 m1^2 m3^2 (Sigma)^2))-2 (Sigma)^2),I-(m1-6 m3)/(6 k m1 m3 (Sigma)^2),(3/m1+1/m3+(12 (Sigma)^4)/m2)/(6 k (Sigma)^2),2 (Sigma)^2,1/(8 k^2 m3^2 (Sigma)^2)+4 (Sigma)^2,(m1+12 m3)/(12 k m1 m3 (Sigma)^2)},{-I-(m1-6 m3)/(6 k m1 m3 (Sigma)^2),1/(Sigma)^2,1/(2 (Sigma)^2),0,0,1/(Sigma)^2},{(3/m1+1/m3+(12 (Sigma)^4)/m2)/(6 k (Sigma)^2),1/(2 (Sigma)^2),1/(Sigma)^2+(4 (Sigma)^2)/(k^2 m2^2),I+(4 (Sigma)^2)/(k m2),(1/m3+(16 (Sigma)^4)/m2)/(4 k (Sigma)^2),1/(Sigma)^2},{2 (Sigma)^2,0,-I+(4 (Sigma)^2)/(k m2),4 (Sigma)^2,4 (Sigma)^2,0},{1/(8 k^2 m3^2 (Sigma)^2)+4 (Sigma)^2,0,(1/m3+(16 (Sigma)^4)/m2)/(4 k (Sigma)^2),4 (Sigma)^2,1/(8 k^2 m3^2 (Sigma)^2)+6 (Sigma)^2,I+1/(4 k m3 (Sigma)^2)},{(m1+12 m3)/(12 k m1 m3 (Sigma)^2),1/(Sigma)^2,1/(Sigma)^2,0,-I+1/(4 k m3 (Sigma)^2),3/(2 (Sigma)^2)}}));
(*The first eigenvalue is 0*)
s((1))
(*Maximum and Minimum of second eigenvalue*)
NMinimize(s((2)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
NMaximize(s((2)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
(*Maximum and Minimum of third eigenvalue*)
NMinimize(s((3)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
NMaximize(s((3)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
(*Maximum and Minimum of fourth eigenvalue*)
NMinimize(s((4)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
NMaximize(s((4)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
(*Maximum and Minimum of fifth eigenvalue*)
NMinimize(s((5)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
NMaximize(s((5)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
(*Maximum and Minimum of sixth eigenvalue*)
NMinimize(s((6)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
NMaximize(s((6)),m1>0&&m2>0&&m3>0&&k>0&&(Sigma)>0,{m1,m2,m3,m3,k,(Sigma)})
```

I have especially tried this method because of the following reasoning:

**If some eigenvalues become purely negative, their maximum and minimum values will also be negative and, in this situation, the matrix defined above will not be a semi-defined matrix.**

Then, in my code, I find that the second eigenvalue (`s((2))`

) therefore has a negative maximum and a minimum, which proves that the matrix has negative eigenvalues and is therefore a non-semi-defined matrix. But the problem is that Mathematica displays the following error at the output of Max and Min calculation.

```
minimize::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. >>
```

So, I have three questions

**How to remove the above error?**
**Is my methodology / reasoning correct for testing the positive semi-finiteness of the matrix?**
**Do you have any other suggestions to prove the semi-definitive positive nature of the aforementioned matrix?**