The odd power of copositive matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:05:19Z http://mathoverflow.net/feeds/question/68678 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68678/the-odd-power-of-copositive-matrix The odd power of copositive matrix Sunni 2011-06-23T23:56:38Z 2011-06-24T02:38:12Z <p>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.</p> <p>Any reference is appreciated</p> http://mathoverflow.net/questions/68678/the-odd-power-of-copositive-matrix/68686#68686 Answer by Gjergji Zaimi for The odd power of copositive matrix Gjergji Zaimi 2011-06-24T02:07:36Z 2011-06-24T02:07:36Z <p>A counter-example is given by $$\pmatrix{.6,0,0,.6,1}\pmatrix{ 1 &amp; -1 &amp; 1 &amp; 1 &amp; -1 \\ -1 &amp; 1 &amp; -1 &amp; 1 &amp; 1 \\ 1 &amp; -1 &amp; 1 &amp; -1 &amp; 1\\ 1 &amp; 1 &amp; -1 &amp; 1 &amp; -1\\ -1 &amp; 1 &amp; 1 &amp; -1 &amp; 1} ^3 \pmatrix{.6 \\ 0 \\ 0\\ .6 \\1}=-0.44.$$</p> <p>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.</p> http://mathoverflow.net/questions/68678/the-odd-power-of-copositive-matrix/68690#68690 Answer by suVRit for The odd power of copositive matrix suVRit 2011-06-24T02:24:27Z 2011-06-24T02:38:12Z <p>My preliminary experiments show that the answer is <em>no</em>. Here is why.</p> <p>In the paper <a href="http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol17_pp09-20.pdf" rel="nofollow">Constructing copositive matrices from interior matrices</a>, the following matrix (from Horn's quadratic form) is mentioned to be copositive:</p> <p>$$A=\begin{bmatrix} 1 &amp; -1 &amp; 1 &amp; 1 &amp; -1\\ -1 &amp; 1 &amp; -1 &amp; 1 &amp; 1\\ 1 &amp; -1 &amp; 1 &amp; -1 &amp; 1\\ 1 &amp; 1 &amp; -1 &amp; 1 &amp; -1\\ -1 &amp; 1 &amp; 1 &amp; -1 &amp; 1\\ \end{bmatrix}$$</p> <p>Now plug this matrix into Matlab, and search to see if $x^TA^3x &lt; 0$ occurs for a nonnegative $x$. Here is one particular value:</p> <p>$$x = \begin{pmatrix} 4\\ 6\\ 3\\ 0\\ 0\\ \end{pmatrix}$$ because with this choice of $x$, we obtain $x^TA^3x = -11$</p>