Quadratic Form
Definition
$V$ is called a $n$dimensional vector space. For a given constant $a_{ij} \in \mathbb{R}(\text{or } \mathbb{C})$, the following second order homogeneous function $A : V \to \mathbb{R}(\text{or } \mathbb{C})$ is called a quadratic form.
$$ A(\mathbf{x}) := \sum\limits_{i,j=1}^{n} a_{ij}x_{i}x_{j},\qquad (a_{ij} = a_{ji}) $$
Here, $\mathbf{x} = \begin{bmatrix} x_{1} & \cdots & x_{n} \end{bmatrix}^{T}$ holds. The term $i \ne j$ for $a_{ij}x_{i}x_{j}$ is called the cross product terms.
Explanation
According to the definition, $A(\lambda \mathbf{x}) = \lambda^{2} A(\mathbf{x})$ holds.
Matrix Form
Let $A$ be a $n\times n$ symmetric matrix $A = \begin{bmatrix} a_{ij} \end{bmatrix}$. The quadratic form of matrix $A$, denoted by $Q_{A}(\mathbf{x})$, is called quadratic form associated with A.
$$ Q_{A}(\mathbf{x}) = \mathbf{x}^{T}A\mathbf{x} =\begin{bmatrix} x_{1} & \cdots &x_{n} \end{bmatrix} \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix}=\sum \limits _{i=1} ^{n}\sum \limits _{j=1} ^{n}a_{ij}x_{i}x_{j} $$
For example, the quadratic form on $\mathbb{R}^{2}$ is as follows.
$$ \begin{align*} & a_{11}^{\ }x_{1}^{2} + a_{22}^{\ }x_{2}^{2} + a_{12}^{\ }x_{1}^{\ }x_{2}^{\ } + a_{21}^{\ }x_{2}^{\ }x_{1}^{\ } \\ =&\ a_{11}^{\ }x_{1}^{2} + a_{22}^{\ }x_{2}^{2} + 2a_{12}^{\ }x_{1}^{\ }x_{2}^{\ } \end{align*} $$
The quadratic form on $\mathbb{R}^{3}$ is as follows.
$$ a_{11}^{\ }x_{1}^{2} + a_{22}^{\ }x_{2}^{2} + a_{33}^{\ }x_{3}^{2} + 2a_{12}^{\ }x_{1}^{\ }x_{2}^{\ } + 2a_{13}^{\ }x_{1}^{\ }x_{3}^{\ } + 2a_{23}^{\ }x_{2}^{\ }x_{3}^{\ } $$
To avoid repetition, it is common to combine cross terms as shown above. The quadratic form can be represented by the properties of matrix inner product as follows. For real and complex numbers, respectively:
$$ \begin{align*} Q_{A}(\mathbf{x}) &= \mathbf{x}^{T} A \mathbf{x} = \mathbf{x} \cdot A\mathbf{x} = A\mathbf{x} \cdot \mathbf{x} = \left< A\mathbf{x}, \mathbf{x}\right> = \left< \mathbf{x}, A \mathbf{x} \right> \\ Q_{A}(\mathbf{x}) &= \mathbf{x}^{\ast} A \mathbf{x} = \mathbf{x} \cdot A\mathbf{x} = A\mathbf{x} \cdot \mathbf{x} = \left< A\mathbf{x}, \mathbf{x}\right> = \left< \mathbf{x}, A \mathbf{x} \right> \end{align*} $$
If $A$ is a diagonal matrix, then since $a_{ij}=0 (i \ne j)$ holds, the quadratic form $Q_{A}(\mathbf{x})$ does not have cross terms.
$$ Q_{A}(\mathbf{x}) = \mathbf{x}^{T}A\mathbf{x} =\begin{bmatrix} x_{1} & \cdots &x_{n} \end{bmatrix} \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix}\begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix}=\sum \limits _{i=1}^{n} a_{ii}x_{i}^{2} $$