Why is positive semi-definiteness for matrices analgous to positiveness for numbers?

Question

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?

Answer 1

I see it as follows.

Assuming here that we are working over real numbers.

Now what is a $1\times 1$ symmetric matrix. It is simply $[b]$ where $b$ is a real number. Let us call this matrix $B$. Now $B$ being positive semidefinite means $$x^t Bx \geq 0 $$ for all $ x \in \mathbb{R} $

This will happen if and only if $b\geq 0$ which conveys nothing but positiveness of real number $b$.

So it makes sense to me. In short think of real numbers as $1\times 1$ matrix. What you have written follows.