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Prove an identity for vectors in Cn

by RoRi

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

    \[ \lVert A + B \rVert^2 = \lVert A \rVert^2 + \lVert B \rVert^2 + A \cdot B + \overline{A \cdot B}. \]

Proof. We can compute, noting that B \cdot A = \overline{A \cdot B} by Theorem 12.11(a),

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

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