While I was reading a paper, I came across the following statement:
Given a semidefinite matrix $G$ with proper dimensions, the $G$-norm of $x$ is $\|x\|_G = \sqrt{x^TGx}$.
When I tried to check the norm properties, I cannot see why if $\|x\|_G = 0$ implies that $x = 0$. At least, to my understanding, I can find $x \neq 0$ such that $\|x\|_G = 0$.