A question on a nonnegative quadratic form

Question

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$.

Could anyone help me prove whether the following quadratic form positive semi-definite?

\begin{equation*} \begin{split} I(x,y,z)=&(a^2+4b^2+4c^2)a^2x^2+(4a^2+b^2+4c^2)b^2y^2\\ +&(4a^2+4b^2+c^2)c^2z^2\\ -&2ab(a^2+b^2+c^2+3ab)xy\\ +&2ac(a^2+b^2+c^2+3ac)xz\\ +&2bc(a^2+b^2+c^2+3bc)yz. \end{split} \end{equation*}

Answer 1

Well, I showed you how to try to attack these problems a couple of days ago, and you can use exactly the same strategy here.

A problem on positive semi-definite quadratic forms/matrices

Once again using MATLAB Toolbox YALMIP to compute a sum-of-squares certificate. Ordering of the variables is not required, but the equality appears to be necessary to exploit

sdpvar a b
c = -a-b
I=(a^2+4*b^2+4*c^2)*a^2*x^2+(4*a^2+b^2+4*c^2)*b^2*y^2+(4*a^2+4*b^2+c^2)*c^2*z^2-2*a*b* (a^2+b^2+c^2+3*a*b)*x*y+2*a*c*(a^2+b^2+c^2+3*a*c)*x*z+2*b*c*(a^2+b^2+c^2+3*b*c)*y*z;
% Floating-point decomposition
[~,v,Q] = solvesos(sos(I))
sdisplay(v{1})
Q{1}
% Integer decomposition
Q = intvar(7,7);
Match = coefficients(I-v{1}'*Q*v{1},[a b x y z])==0;
optimize([Match, Q >=0])
value(Q)

>> sdisplay(v{1})

ans = 

'z*b^2'
'z*a*b'
'z*a^2'
'y*b^2'
'y*a*b'
'x*a*b'
'x*a^2'

>> value(Q)

ans =

 5     6     1     1     1    -2     0
 6    12     6     1    -1    -1     1
 1     6     5     0    -2     1     1
 1     1     0     5     4    -2    -2
 1    -1    -2     4     8    -3    -2
-2    -1     1    -2    -3     8     4
 0     1     1    -2    -2     4     5