# When is the difference of the “1” matrix and a positive semidefinite matrix positive semidefinite?

Consider a positive semidefinite matrix $A\in M_n(\mathbb{C})$. Let $E$ denote the matrix in $M_n(\mathbb{C})$ all of whose entries are $1$.

What are some natural sufficient conditions for $E-A$ to be positive semidefinite?

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Let $P=\frac{1}{n}E$, so $P$ is a rank-one orthogonal projection, and it suffices to consider when $P-A$ is positive semidefinite. If $0\leq A\leq P$, then $A^{1/k}\leq P$ for all $k\geq 1$. Letting $k\to \infty$, if $Q$ is the projection onto the range of $A$, then $Q\leq P$. Since $P$ has rank 1, this means that $Q=P$ or $Q=0$. This implies that $A=cP$ for some $c\in[0,1]$. This necessary condition is also sufficient.