The odd power of copositive matrix - MathOverflow

The odd power of copositive matrix

2011-06-23T23:56:38Z

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

Answer by Gjergji Zaimi for The odd power of copositive matrix

2011-06-24T02:07:36Z

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.

Answer by suVRit for The odd power of copositive matrix

2011-06-24T02:24:27Z

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 $x^TA^3x = -11$