If you have two matrices, $A$ and $B$, that are symmetric and square, you can define an ordering on your matrices as follows:

Loewner Order

$A \succeq B \iff (A-B) \succeq 0$. In other words, the difference between matrices $A-B$ is positive semi-definite.

If $A \succeq B$ and $A$ and $B$ are symmetric and square, we know the following:

1. Min-Max Ordering: The $k^{th}$ smallest eigenvalue of $B$ is less than or equal to the $k^{th}$ smallest eigenvalue of $A$.

   $$ \lambda_k(B) \leq \lambda_k(A), \forall k $$

2. Determinant and Trace: It also holds that $det(A) \geq det(B)$ and $tr(A) \geq tr(B)$.