If $a$ and $b$ are two numbers on the real line, we compare $a$ and $b$ by knowing which of them comes first as we move from $-\infty$ to $\infty$ on the real line.
However when $A$ and $B$ are matrices, the comparison is through definiteness. We say $A \succ B$ iff $A-B$ is positive definite. Positive definiteness of $A$ means $x^TAx>0\ \forall x$; essentially the function $f(x)=x^TAx$ takes the form of a bowl with its base at origin.
How does this "bowl" help in comparing two matrices? What is the intuition behind using definiteness in matrices for ordering?
Best Answer
Where did you get your definition from?
I think you might be confusing the notation here. $A \succ B$ simply is a notation used to indicate that $A$ and $B$ are symmetric matrices of same size such that $A - B$ is positive definite. The reason such a notation is used is because there are certain (monotonic) properties these matrices satisfy. For example, if $A \succ B$ then $\lambda(A) > \lambda(B)$.
Now, there is another similar relation which says $A \succeq B$ which is used to express that $A$ and $B$ are symmetric matrices of same size such that $A - B$ is positive semi-definite. Now, this also has similar (monotonic) properties as the relation above.
However, one thing about the $\succeq$ relation is that it does define a partial order over all symmetric matrices (You can take this as an exercise, it is quite easy to show). This is called Lowner ordering. In this sense, one can say that we are comparing $A$ with $B$. But again, you have to understand that it is not for arbitrary matrices, only for symmetric matrices. Also, this ordering is partial and not total.