Suppose there is a square matrix $A$ and a positive semi-definite matrix $X\in\Re^n$, such that \begin{equation} \mathrm{trace}(AX)\leq0 \end{equation} Is there any ways I could do the rank one decomposition of matrix $X$, such that for $r\leq n$, \begin{equation} X=\sum_{k=1}^{r} x_kx_k^\top \end{equation} and keep the inquality constraints \begin{equation} x_k^\top A x_k = \mathrm{trace}(A(x_kx_k^\top))\leq0 \qquad \forall k \end{equation} Or at least hold for one $x_{k}$?

Many thanks!


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