summation - Differences In Explanations of Quadratic Functions/Forms

My textbook explains quadratic functions as follows:

Quadratic functions are functions $g: \mathbb{R^n} \to \mathbb{R}$ that have the form

$$g(h_1, \dots h_n) = \sum_{i, j = 1}^n a_{ij} h_i h_j$$

for an $n \times n$ matrix $[a_{ij}]$.

Wikipedia defines quadratic functions as follows:

$$q(x_1, \dots, x_n) = \sum_{i = 1}^n \sum_{j = 1}^n a_{ij} x_i x_j$$

I have some questions regarding these definitions:

1. There are two forms of summation notation used: $\sum_{i, j = 1}^n$ and $\sum_{i = 1}^n \sum_{j = 1}^n$. Now, as I understand it, these are not necessarily equivalent notations, since, with $\sum_{i, j = 1}^n$, we always have $i= j$, whereas with $\sum_{i = 1}^n \sum_{j = 1}^n$, we can have $i \not= j$, since one summation, say, $\sum_{j = 1}^n$, is summed over while the other, $\sum_{i = 1}^n$, is held constant at $i = c \in [1, n]$, and only incremented by one once the summation over all values of $j \in [1, n]$ is complete. It seems to me like both of these can't be correct, so what's going on here? Am I misunderstanding something? Since $[a_{ij}]$ is an $n \times n$ matrix, it seems to me like the double-summation $\sum_{i = 1}^n \sum_{j = 1}^n$ is the one we would be looking for?




2. How do these above definitions of quadratic functions/forms relate to the commonly-known quadratic function/equation $f(x) = ax^2 + bx + c$?

I would appreciate it if people could please take the time to clarify this.

Answer

A quadratic form on a vector space should be thought of as a homogeneous quadratic function. Here homogeneous means that every monomial must have the same degree, and quadratic means that it must be degree 2. For example $f(x, y, z) = xy + 3yz - x^2$ is a homogeneous quadratic in three variables, but $g(x, y, z) = x^2 + y + y^2 - 7$ is not, because of the degree-1 term $y$, and the degree-0 term $-7$. Hopefully you can see how the definition given on the wikipedia page allows all functions like $f$, while not including ones like $g$.

As for your second question, the only quadratic forms in one variable are $f(x) = ax^2$ for some $a \in \mathbb{R}$. So there is perhaps not so much relation to quadratic polynomials in a single variable.

If you want a classic example of a quadratic form, the norm squared on a vector space is a quadratic form:

$$ f(x_1, \ldots, x_n) = \lvert (x_1, \ldots, x_n) \rvert^2 = x_1^2 + \cdots + x_n^2$$