Is it true that any inner product $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{C}$ is given by $$\langle \boldsymbol{x}, \boldsymbol{y} \rangle = \boldsymbol{x}^T \boldsymbol{M} \boldsymbol{\bar{y}}$$ where $\boldsymbol{M}$ is a positive definite, hermitian matrix, $\boldsymbol{x}, \boldsymbol{y} \in V$?
Is it true that any inner product is also given by $$\langle \boldsymbol{x}, \boldsymbol{y} \rangle = \boldsymbol{y}^* \boldsymbol{M} \boldsymbol{x}$$ where $\boldsymbol{x}^*$ denotes the conjugate transpose of $\boldsymbol{x}$?