Home » Blog » Prove another identity for vectors in Cn

Prove 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:

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

Proof. From the previous exercise (Section 12.17, #3) we have the identity

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

This also give us

    \begin{align*} \lVert A - B \rVert &= \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}. \end{align*}

Therefore,

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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):