product of a vector times an asymmetric matrix vs a symmetric matrix

Question

As a school assignment, im currently finding the gradient for different functions, however of the functions are the same (where x is a vector, A is a matrix, b a vector, c a scalar): $$f(x)=x^{T}Ax+b^{T}x+c$$ Except that A is not symmetric in one function, and symmetric in the other, however this shouldn't make a difference since $A \cdot x$ simply provides a new vector, or am i completely wrong here?

Answer 1

Do out the algebra. In one case, there'll be a simplification that arises from $a_{jk} = a_{kj}$; in the other, that simplification won't be there. For instance, I might give you a function of a matrix $A$: $$ h(A) = {A + A^t} $$ and another function $$ g(A) = 2A $$ These two are the same if $A$ is symmetric, but different if not. In much the same way, the formula you derive will be "more simplifiable" if $A$ is symmetric than if it isn't. Imagine, for instance, that there's a term like $a_{jk} - a_{kj}$ -- for symmetric matrices, that disappears.