If $A$ is copositive, what about $A^3$? Is it also copositive? More generally,
my question is whether the odd power of a copositive matrix is still copositive.


Any reference is appreciated



The odd power of copositive matrix
A counter-example is given by
$$\pmatrix{.6,0,0,.6,1}\pmatrix{
1 & -1 & 1 & 1 & -1 \\\
-1 & 1 & -1 & 1 & 1 \\\
1 & -1 & 1 & -1 & 1\\\
1 & 1 & -1 & 1 & -1\\\
-1 & 1 & 1 & -1 & 1} ^3 \pmatrix{.6 \\\ 0 \\\ 0\\\ .6 \\ 1}=-0.44.$$


while the matrix in the centre is actually copositive. It is easy to check that there are no counter-examples for $2\times 2$ matrices.


