Quadratic Form

These are representations of expressions in the form: $x_1^2+\cdots +x_n^2+{\sum }_{i=1}^n(x_i{x}_j|i\ne j)$

They can represent Matrixes obtained from Polarization Identity

Definition

• Let $x\in {\mathbb{R}}^n$, A function $x^{T}Ax=f(x_1,\dots ,x_n)={\sum }_{i=1}^n{\sum }_{j=1}^n{a}_{ij}{x}_i{x}_j$ creates quadratic form
• Let $x\in {\mathbb{C}}^n$, A function $x^{T}Ax=f(x_1,\dots ,x_n)={\sum }_{i=1}^n{\sum }_{j=1}^n{a}_{ij}\overline{x_i}{x}_j$ creates quadratic form

Theorems

• Matrix Representation of Quadratic Form