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?
Best Answer
Any matrix A can be written as sum of $(A+A^{T})/2$ and $(A-A^{T})/2$. You can verify that the quadratic form of second term (i.e; $x^{T}(A-A^{T})x$) turns out to be zero (Try to evaluate the second term and check for yourself).
Hence, we can assume that we begin with a symmetric matrix. If not, it is anyway possible to convert to an equivalent quadratic form with a symmetric matrix.