$A \lambda A^{T} $ (quadratic form?) is used with matrices to check definiteness. What about with vectors? If I see conditions such as $\bar{x} > 0$, how can I know whether it means $\bar{x}_{i} > 0 \quad \forall i \in I$ or $\bar{x}:=\bar{x} \lambda \bar{x}^{T}>0$?
I am trying to understand the conditions in the standard form integer optimization problem where I see such cases:
$ \text{minimize }\ \ \bar{c}^{T}\bar{x} + \bar{d}^{T}\bar{y}$
so that $A\bar{x} + B\bar{y} = b$ $\bar{x}, \bar{y} \geq 0$ and $\bar{y}$ is continuous.