Can A>B Be Determined by A-B>0 for Matrices?

[matqkks]

Can we compare matrices?
If A-B>0 is positive definite, can we say A>B?

 

[jbunniii]

Can we compare matrices?
If A-B>0 is positive definite, can we say A>B?

Yes, this is valid notation. This definition of $>$ gives us a strict partial ordering on the set of $N\times N$ matrices. Similarly, you can define $A \geq B$ if $A - B$ is positive semidefinite.

Note that if $A > B$ and $B > C$, then $A - B$ and $B - C$ are positive definite, and $A - C = (A - B) + (B - C)$. As the sum of two positive definite matrices is positive definite, this shows that the transitivity axiom of (partial or strict) ordering is satisfied. The other axioms are even easier to check.