I have a vector $x = [y^\intercal_1, y^\intercal_2, \dots, y^\intercal_n]^\intercal \in \mathbb{R}^d$, where $y_1 \in \mathbb{R}^{d_1}, \dots, y_n \in \mathbb{R}^{d_n}$ are column vectors. I want to find an upper bound for the outer product $x x^\intercal$ in the following sense: find a matrix $B$ such that
$$x x^\intercal = \begin{bmatrix} y_1 y_1^\intercal & \dots y_1 y_n^\intercal\\ \vdots & \quad \vdots\\ y_n y_1^\intercal & \dots y_n y_n^\intercal\end{bmatrix} \preceq B,$$ which means that $B - xx^\intercal$ is positive semi-definite. Are there general methods for finding such bounds? I know that I cannot construct $B$ by simply bounding each element of $xx^\intercal$ because in that case, any matrix with all positive elements would be PSD.