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Prove yet another identity for vectors in Cn

by RoRi

Let A, B \in \mathbb{C}^n be any two vectors. Prove that we have the following identity,

    \[ \lVert A + B \rVert^2 + \lVert A - B \rVert^2 = 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \]

Proof. Using our computations of \lVert A + B \rVert^2 and \lVert A - B \rVert^2 in the previous exercise (Section 12.17, Exercise #4) we have

    \begin{align*} \lVert A + B \rVert^2 + \lVert A - B \rVert^2 &= \lVert A \rVert^2 + \lVert B \rVert^2 + A \cdot B + \overline{A \cdot B} + \lVert A \rVert^2 + \lVert B \rVert^2 - A \cdot B - \overline{A \cdot B} \\ &= 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \qquad \blacksquare \end{align*}

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