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Prove some facts about vectors in Cn

  1. Prove that for A, B \in \mathbb{C}^n we have

        \[ \overline{A \cdot B} + A \cdot B \in \mathbb{R}. \]

  2. For non-zero vectors A,B \in \mathbb{C}^n prove that

        \[ -2 \leq \frac{A \cdot B + \overline{A \cdot B}}{\lVert A \rVert \lVert B \rVert} \leq 2. \]


  1. Proof. Let A \cdot B = x + iy for x, y \in \mathbb{R}.

        \[ A \cdot B + \overline{A \cdot B} = (x+iy) + (x-iy) = 2x \in \mathbb{R}. \qquad \blacksquare \]

  2. Proof. Let A \cdot B = x + iy for x,y \in \mathbb{R}. Then,

        \[ A \cdot B + \overline{A \cdot B} = 2x \]

    and

        \[ \lVert A \rVert \lVert B \rVert = \left( A \cdot A \right)^{\frac{1}{2}} \left( B \cdot B \right)^{\frac{1}{2}} = \left( (A \cdot B)(A \cdot B) \right)^{\frac{1}{2}} = \sqrt{x^2+y^2}. \]

    Therefore,

        \begin{align*}  \frac{A \cdot B + \overline{A \cdot B}}{\lVert A \rVert \lVert B \rVert} &= \frac{2x}{\sqrt{x^2+y^2}} \\  &\leq \frac{2x}{\sqrt{x^2}} \\  &= \frac{2x}{|x|} \\  &= \pm 2. \end{align*}

    Therefore,

        \[ -2 \leq \frac{A \cdot B + \overline{A \cdot B}}{\lVert A \rVert \lVert B \rVert} \leq 2. \qquad \blacksquare \]

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