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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?

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    Perhaps one answer is just that it's a convenient notation. Instead of saying "$A$ is positive definite", we can say $A \succ 0$. Instead of saying, "$A - B$ is positive definite", we can say $A \succ B$. This notation reminds us that $\succeq$ is indeed a partial order on $\mathbb R^{n \times n}$.2012-11-17
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    In other words, it is not that "*the* comparison [of matrices] is through definiteness", but only that definiteness is *one way* to compare matrices. There are certainly other ways to define (partial) orderings on matrices, though those aren't as common. The notation $A\succeq B$ itself doesn't mean much more than "$A$ succeeds $B$ in whatever partial order I had told you earlier we were using".2012-11-17

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