\part{Quadratic forms}

\chapter{Quadratic forms}
\[ q(\mathbf{x}) =\mathbf{x}^{\intercal}  \mathbf{A} \mathbf{x} \]

\[ q(\mathbf{x}) = \sum_{i=1}^n \sum_{j=1}^n \mathbf{A}_{ij} x_i x_j \]


\begin{propostition}
For any quadratic form $q$, there exists a unique symmetric matrix $\mathbf{A}$ such that $q$ is representable as such.
\[ q(\mathbf{x}) =\mathbf{x}^{\intercal}  \mathbf{A} \mathbf{x} \]
\end{propostition}


\begin{propostition}
Let $ q(\mathbf{x}) =\mathbf{x}^{\intercal}  \mathbf{A} \mathbf{x}$ be a quadratic form, then  $\frac{\mathbf{A} + \mathbf{A}^{\intercal}}{2}$ is a symmetric matrix that represents the same quadratic form
\end{propostition}


