I'm studying an undergraduate level geometry book and was studying about inner products when I got a bit confused. I've tried to find other answers here and elsewhere, but none of the answers were exactly intuitive and so it was hard for me to understand, and so decided to ask my own question.
According to the book, one of the properties of the inner product between two vectors is that it must be positive definite. To borrow the exact words:
An inner product on $\Bbb{R}^n$ is a function $\langle\ \cdot\ ,\ \cdot\ \rangle: \Bbb{R}^n \times \Bbb{R}^n \rightarrow \Bbb{R}$ on two vector variables that satisfies the following properties:
Positive definiteness: The necessary and sufficient condition for $\langle\mathbf{a}, \mathbf{a} \rangle \ge 0$ and $\langle\mathbf{a}, \mathbf{a}\rangle = 0$ is $\mathbf{a} = \mathbf{0}$.
Commutativity: $\langle \mathbf{a}, \mathbf{b} \rangle = \langle \mathbf{b}, \mathbf{a} \rangle$
Linear on the first argument: $\langle \mathbf{a}_1 + \mathbf{a}_2, \mathbf{b} \rangle = \langle \mathbf{a}_1, \mathbf{b} \rangle + \langle \mathbf{a}_2, \mathbf{b} \rangle$ and $\langle \alpha \mathbf{a}, \mathbf{b} \rangle = \alpha \langle \mathbf{a}, \mathbf{b} \rangle = \langle \mathbf{a}, \alpha \mathbf{b} \rangle$
I'm having trouble understanding the positive definiteness. Why is that so? What is the geometric meaning of an inner product having to be positive definite? In fact, I've never even heard of this before when I was studying linear algebra. I merely learned that the inner product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:
$$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^n a_ib_i$$
One Reddit answer brought up the concept of "distance" and that if an inner product is not positive definite then we cannot define distance between two vectors, but I'm having trouble understanding that as well.
Also, I thought that positive definiteness did not include equality (i.e. $\ge$) and rather positive semi-definiteness is the one that included equality.
Would anyone be able to shed some light on this concept? Thanks in advance.
Best Answer
Yes, it is part of the definition of inner product that we always have $\langle v,v\rangle\geqslant0$. That's because that allows us to define a norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ and from that norm we get a distance: the distance from $v$ to $w$ is $\lVert v-w\rVert$.
But I don't think I've ever seen “Positive definiteness” as a name for this property. It has nothing to do with positive definite matrices.