Why does the inner product of two vectors have to be positive definite?

Question

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:

1. 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 $\mathbf{a} = \mathbf{0}$.

2. Commutativity: $\langle \mathbf{a}, \mathbf{b} \rangle = \langle \mathbf{b}, \mathbf{a} \rangle$

3. 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.

Answer 1

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.

Answer 2

In addition to Jose's answer, if $A$ is a positive definite matrix, then $⟨x,y⟩:=x^TAy$ defines an inner product. The norm and the distance induced by an inner product give a metric $d(x,y)$, which satisfies $d(x,y)=0$ if and only if $x=y$ because the inner product is "positive definite". And this property is very natural for any metric.

Answer 3

Your confusion stems from this:

I merely learned that the inner product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:

$$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^n a_ib_i$$

This is the usual definition of inner product in $\Bbb R^n$. In more advanced classes, we learn that there are other possible definitions of an inner product on a vector space. But if we want to call $\left<x,y\right>$ an inner product, it has to obey certain conditions, one of which is that $\left<x,x\right>\ge 0$, with $\left<x,x\right>=0$ if and only if $x=0$.

By the way, the definition of positive definiteness that you give in your question is garbled. It should be something like:

Positive definiteness: $\langle\mathbf{a}, \mathbf{a} \rangle \ge 0$ for all $\mathbf{a}$, and the necessary and sufficient condition for $\langle\mathbf{a}, \mathbf{a}\rangle = 0$ is $\mathbf{a} = \mathbf{0}$.