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
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 


A counterexample 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 counterexamples for $2\times 2$ matrices. 


My preliminary experiments show that the answer is no. Here is why. In the paper Constructing copositive matrices from interior matrices, the following matrix (from Horn's quadratic form) is mentioned to be copositive: $$ A=\begin{bmatrix} 1 & 1 & 1 & 1 & 1\\\\ 1 & 1 & 1 & 1 & 1\\\\ 1 & 1 & 1 & 1 & 1\\\\ 1 & 1 & 1 & 1 & 1\\\\ 1 & 1 & 1 & 1 & 1\\\\ \end{bmatrix} $$ Now plug this matrix into Matlab, and search to see if $x^TA^3x < 0$ occurs for a nonnegative $x$. Here is one particular value: $$x = \begin{pmatrix} 4\\\\ 6\\\\ 3\\\\ 0\\\\ 0\\\\ \end{pmatrix} $$ because with this choice of $x$, we obtain $x^TA^3x = 11$ 

