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
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Sign up to join this communityIf $A$ 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 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.
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 $x^TA^3x = -11$