Entry Bounds for Positive Semidefinite Matrices

Let $k\ge1$ be a natural number and let $P$ be a positive semidefinite real $k\times k$ matrix with entries $P_{ij}$.

1. $P_{ii}\ge0$ for every $i$, and for all $i,j$, with the nonnegative square root,

2. If $Q$ is a symmetric real $k\times k$ matrix with $P\preceq Q$ in the semidefinite order, then $P_{ii}\le Q_{ii}$ for every $i$ and