Is it always true for vector and matrix multiplication?

Question

I have seen that $$\frac{\partial \textbf{x}^T\textbf{B}\textbf{x}}{\partial \textbf{x}}=\textbf{x}^T(\textbf{B}+\textbf{B}^T)$$ and also $$\frac{\partial \textbf{x}^T\textbf{B}\textbf{x}}{\partial \textbf{x}}=(\textbf{B}+\textbf{B}^T)\textbf{x}$$ which suggests that $\textbf{x}^T\textbf{C}=\textbf{Cx}$? Is it always true where $\textbf{C=B+B}^T$? Any help in this regard will be much appreciated. Thanks in advance

Answer 1

These are two different convention. The first equation is the row convention and the second equation is the column convention. We shouldn't use two convention simultaenously.

$x^TC$ is a row vector while $Cx$ is a column vector. They can't be equal.

Answer 2

Do not confuse. Different authors have proposed different definitions for derivative of a scalar relative to a vector. indeed you have considered two different definitions equal.If you see H.Khalil book of nonlinear control he has used the first and F.Lewis in optimal control has used the second.Logically you can not take two different definitions of a same things equal. It is better to put a triangleover the equal sign