# Rank one decomposition of a positive semi-definite matrix with inequality trace constraints

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

Many thanks!