I am looking for a matrix analogue of comparison theorem. Let's say we consider the two ODEs: $$\dot{X} = A(t)X(t), \ X(0) = \Xi$$ and $$\dot{Y} = B(t)Y(t), \ Y(0) = \Xi.$$
$A$ and $B$ are continuous (and perhaps smooth). I have two notions of order in hand. Do we have
If $A - B$ is positive definite for all $t$, can we say $X - Y$ is positive definite?
If each entry of $A$ is greater than or equal to $B$, i.e., $a_{ij} \geq b_{ij}$ for each $ij$ entry, can we say the same for the solution, i.e., does $x_{ij} \geq y_{ij}$?
I have read a lot on comparison theorem on matrix Riccati equation, but I can't find some helpful references on this simple type equation.