Matrix Representation of Quadratic Form

Theorem

Given $x\in {\mathbb{R}}^n$:

• $f$ as the Quadratic Form $f(x)={\sum }_{i=1}^n{a}_{ij}{x}_i{x}_j$
• With $A:a_{ij}$
• Then, $x^{T}Ax=f(x)$ Given $x\in {\mathbb{C}}^n$:
• $f$ as the Quadratic Form $f(x)={\sum }_{i=1}^n{a}_{ij}{\overline{x}}_i{x}_j$
• With $A:a_{ij}$
• Then, $x^{\ast }Ax=f(x)$

For $2\times 2$ Matrix Example

For $\mathbb{R}$

1. With $x\in {\mathbb{R}}^n$
2. With $f$ as a quadratic form $f(x)=a{x}_1^2+b{x}_1{x}_2+c{x}_2^2$

a & \frac{b}{2}\ \frac{b}{2} & c\ \end{array}\right]$$

For $\mathbb{C}$

1. With $x\in {\mathbb{R}}^n$
2. With $f$ as a quadratic form $f(x)=a_{11}|x_1{|}^2+a_{12}{\overline{x}}_1{x}_2+a_{21}{\overline{x}}_2{x}_1+a_{22}|x_2{|}^2$

a_{11} & a {12}\ a{21} & a _{22} \end{array}\right]$$