er bounds.  0.  **The Positive-Semidefinite Case.** Consider an independent family $\{ {{\bm{T}}}_0, \dots, {{\bm{T}}}_n \}$ of random $d \times d$ positive-semidefinite matrices, and define the sum $${{\bm{W}}} := \sum_{i=0}^n {{\bm{T}}}_i.$$ Then $$\label{eqn:psd-case-lower}     {\operatorname{\mathbb{E}}}{\left\Vert { {{\bm{W}}} } \right\Vert}         \