Difference of positive semidefinite matrices where $A_{ij} \geq B_{ij}$?

Question

Let $A, B \in \mathbb R^{n\times n}$ be two positive semidefinite matrices. Assume that $A_{ij}\geq B_{ij}$ for $i,j\in \{1,\dots,n \}$. Does it hold that $C = A - B$ is positive semidefinite?

One possible simplification that I'm also interested in is if $B$ is of rank 1.

Answer 1

The answer is no, even if $B$ has rank $1$. Consider $$A = \begin{bmatrix} 4 & 2 \\ 2 & 1\end{bmatrix} \ge 0, \quad B = \begin{bmatrix} 1 & 1 \\ 1 & 1\end{bmatrix} \ge 0, \quad A-B = \begin{bmatrix} 3 & 1 \\ 1 & 0\end{bmatrix} \not\ge 0.$$

Answer 2

In general the answer is NO.

Example:

Let $A=\begin{pmatrix}3&2\\2&3\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\0&2\end{pmatrix}$

$C= A-B$ has eigenvalues $-1,3$, and is thus not positive semidefinite, although $A,B$ satisfy the conditions as mentioned in the question above.