M=Table[m[i,j],{i,1,4},{j,1,4}];eig=Eigenvalues[M];V=Eigenvectors[M];list2={V[[4,1]]>=0};For[j=1,j≤3,j++,list2=Join[{V[[4,j]]>=0},list2]];For[i=1;list3={eig[[4]]==0},i≤3,i++,list3=Join[{eig[[i]]<=0},list3]];For[i=1;list4={},i≤4,i++,For[j=1,j≤4,j++,list4=Join[{m[i,j]≥0},list4]]]For[i=1;list5={},i≤4,i++,For[j=1,j≤4,j++,list5=Join[{m[i,j]},list5]]]FindInstance[Join[list2,list3,list4],list5,Reals]
Out[]=
$Aborted
In[]:=
TrueQ[V[[4,4]]>=0]
Out[]=
True
In[]:=
list4
Out[]=
{m[4,4]≥0,m[4,3]≥0,m[4,2]≥0,m[4,1]≥0,m[3,4]≥0,m[3,3]≥0,m[3,2]≥0,m[3,1]≥0,m[2,4]≥0,m[2,3]≥0,m[2,2]≥0,m[2,1]≥0,m[1,4]≥0,m[1,3]≥0,m[1,2]≥0,m[1,1]≥0}