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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 $\boldsymbol{x}$?

  • 2
    What are your assumptions on $V$? (i.e. finite dimensional, base field $\Bbb{C}$, etc)2012-11-25
  • 0
    Say $V= \mathbb{C}^2$2012-11-25
  • 1
    **Hint:** Put elements of the standard basis in place of $\bf x$ and $\bf y$.2012-11-25
  • 0
    Since we multiply matrices by vectors, $V$ must be a set of column vectors.2012-11-26

2 Answers 2