I am looking to the review document for linear algebra (Zico Kolter (updated by Chuong Do), Linear Algebra Review and Reference), and the part of the quadratic form (pg17) mentions about an assumption of being symmetric for a matrix in quadratic form. It also includes some declarative equality for that proposed argument.
What is the practical reason to assume that the matrix describing a quadratic form (over $\mathbb{R}$) is symmetric? I also do not get the idea proposed by the argument? Can someone light me about?
The main reason for getting the matrix of a real quadratic form symmetric by replacing the original matrix with its symmetric part ${A+A^T}\over 2$ is that any symmetric matrix is orthogonally diagonalizable and all eigenvalues are real. Then for symmetric $A$ you have some orthogonal matrix $U$ (that is, $U^T=U^{-1}$) and $U^T A U=D$ is real diagonal, and $x^T A x=(Ux)^T(UAU^T)(Ux)=(Ux)^TD(Ux)$ is a sum of squares with real coefficients where the length $\Vert x \Vert$ of the vector $x$ is the same as the length $\Vert Ux\Vert$ of the vector $Ux$.