Definition 0.0.1  The outer product denoted $\mathbf{v}\mathbf{w}^*\in\mathbb{C}^{n\times n}$ of two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}$ is the matrix of products of there elements.
\[\mathbf{v}\mathbf{w}^* = (v_iw_j) \in \mathbb{C}^{n\times n}\]