I need to show this result:
Given the system of ODEs $x'=Ax$, the origin, $0$, is an attractor (equivalently, all the eigenvalues of the real matrix $A$ are negative) if and only if there exists a positive definite quadratic form $q$ such that $Dq(x)\cdot Ax<0$ for all $x\neq 0$ (D is the differential operator, $\cdot$ is the usual inner product).
I have no idea how to start. I tried to do it in the 2x2 case, but expanding the expression gave me no clues. I believe I have to find an expression for $q$ which will imply in the inequality I want.
If anyone could give me some hint, I would be grateful; I ask not to solve it fully, since I could use any development on Linear Algebra I could get.