TITLE: The odd power of copositive matrix
QUESTION [7 upvotes]: 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

REPLY [11 votes]: 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 $x^TA^3x = -11$