Alright, this question is going to be really broad, and that's my fault because I am extremely lost. I appreciate all the help!
As far as I understand, writing a quadratic multivariable function in quadratic form means writing it like this:
$$f(\vec{x})=\vec{x}^TA\vec{x}+\vec{x}^T\vec{b} + c$$
Where $A$ is a symmetric matrix holding the quadratic coefficients and the halves of the cross-coefficients, $b$ is a vector holding the linear coefficients, and $c$ is some constant.
(I may be wrong about the above...)
My friend was trying to explain to me a geometric reason why the gradient of $f(\vec{x})=\vec{x}^TA\vec{x}$, with none of the other terms, is $2A\vec{x}$.
I honestly wasn't understanding what he was saying, but he kept on mentioning spheres and ellipsoids and talking about growing $\vec{x}$ in the direction that "grows the ellipsoid the fastest"...or something like that.
Can anyone explain what he was talking about, as geometrically as possible?
Thanks!