Suppose $Q$ is symmetric and positive semi-definite, then what properties can we know from $\text{sgn}^T(x)Qx$? Especially, is there any relation between $\text{sgn}^T(x)Qx$? and $\text{sgn}^T(x) x$?
Clarify on the notation: $\text{sgn}(x)=1 ,x>0;\; \text{sgn}(x)=-1 ,x<0;\; \text{sgn}(x)=0 ,x=0$.