Arguments (α and β) of the two complex eigenvalues when A^k is a QM-matrix for 1 ≤ k ≤ 8.

Let A be a real square matrix of size n. A is a  QM-matrix if A^k is a QM-matrix for all k≥1. A is a QM-matrix if the sum of the  k-by-k  principal minors is positive (1 ≤ k ≤ n).

Let A be a real square matrix of size 5 with one real eigenvalue and two complex and non real eigenvalues with arguments α and β. A is a  QM-matrix iff for all n≥1 we have 1+2 cos(n α)+2 cos (n β )>0 and 1 + 2 cos (n α ) cos(n β)+ cos(n β) + cos(n α)>0.

The figure represents (white area) the values of α and β satisfying the two previous conditions for n=1,2,...,8. Final conclusion says that such a matrix does not exist if α and β are different and not zero.

Code included:

display([
implicitplot([
cos(beta)+cos(alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[green]),
implicitplot([
cos(beta) + 1 + cos(alpha) + cos(-beta + alpha) + cos(beta + alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[green]),
implicitplot([
cos(2*beta)+cos(2*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[yellow]),
implicitplot([
cos(2*beta) + 1 + cos(2*alpha) + cos(-2*beta + 2*alpha) + cos(2*beta + 2*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[yellow]),
implicitplot([
cos(3*beta)+cos(3*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[red]),
implicitplot([
cos(3*beta) + 1 + cos(3*alpha) + cos(-3*beta + 3*alpha) + cos(3*beta + 3*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[red]),
implicitplot([
cos(4*beta)+cos(4*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[orange]),
implicitplot([
cos(4*beta) + 1 + cos(4*alpha) + cos(-4*beta + 4*alpha) + cos(4*beta + 4*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[orange]),
implicitplot([
cos(5*beta)+cos(5*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[blue]),
implicitplot([
cos(5*beta) + 1 + cos(5*alpha) + cos(-5*beta + 5*alpha) + cos(5*beta + 5*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[blue]),
implicitplot([
cos(6*beta)+cos(6*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[pink]),
implicitplot([
cos(6*beta) + 1 + cos(6*alpha) + cos(-6*beta + 6*alpha) + cos(6*beta + 6*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[pink]),
implicitplot([
cos(7*beta)+cos(7*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[cyan]),
implicitplot([
cos(7*beta) + 1 + cos(7*alpha) + cos(-7*beta + 7*alpha) + cos(7*beta + 7*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[cyan]),
implicitplot([
cos(8*beta)+cos(8*alpha)+1/2<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=blue, gridrefine=2,filledregions=true,coloring=[brown]),
implicitplot([
cos(8*beta) + 1 + cos(8*alpha) + cos(-8*beta + 8*alpha) + cos(8*beta + 8*alpha)<=0],
alpha=-Pi..Pi,beta=-Pi..Pi,color=black, gridrefine=2,filledregions=true,coloring=[brown]),
pointplot([[0,2*Pi/3],[0,-2*Pi/3],[2*Pi/3,0],[-2*Pi/3,0]],color=violet, symbol=solidcircle,symbolsize=8)])