I am reading a textbook which states
for $A, B$ symmetric, $A \leq B$ iff $B-A$ is non-negative definite (equiv. positive semi-definite). This is easily seen to be an ordering.
This reads like the following statement for real numbers, applied to matrices: "$a \leq b$ iff $b-a$ is non-negative."
It's not intuitively clear to me why ordering for matrices is defined using positive definiteness. For instance, we could define a "positive" matrix as one that has all positive entries, but we don't do that. What is special about the $x^TAx \geq 0$ definition for positive semi-definite matrices that allows the notion of positiveness of real numbers to generalize to matrices in a useful manner?