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Vector inequality

Let \mathbb{R}^n be endowed with the usual product and the usual norm. If v = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n then we define \sum v = x_1 +x_2 + \cdots + x_n. Prove that

    \[\left \| v \right \|^2 \left \| w \right \|^2 \geq \left ( v \cdot w \right )^2 + \frac{\left ( \left \| v \right \| \left | \sum w \right | - \left \| w \right \| \left |\sum v \right | \right )^2}{n}\]

Solution ( Robert Tauraso )

We will show the more general inequality

    \[\left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left \| u \right \|^2 \geq \left \| \left ( w, u \right ) v - \left ( v, u \right ) w \right \|^2\]

where u \in \mathbb{R}^{n}. Taking u=(1, 1 , \dots, 1) we get the requested inequality. If v, w are linearly dependent then \left \| v \right \|^2 \left \| w \right \|^2 = \left ( v \cdot w \right )^2 the inequality holds. We assume now that v and w are linearly independent. Then u = \alpha v + \beta w + z where \alpha, \beta \in \mathbb{R} and z \perp vz \perp w. Moreover,

    \[\left\{\begin{matrix} \left ( v, u \right ) & = & \alpha \left \| v \right \|^2 + \beta \left ( v, w \right ) \\\\ \left ( w, u \right ) & = & \alpha \left ( v, w \right ) + \beta \left \| w \right \|^2 \end{matrix}\right.\]

and by solving the linear system we find

    \[\alpha = \frac{\left ( v, u \right ) \left \| w \right \|^2 - \left ( w, u \right ) \left ( v, w \right )}{\left \| v \right \|^2 \left \| w \right \|^2 - \left ( v, w \right )^2} \quad , \quad \beta = \frac{\left ( w, u \right )\left \| u \right \|^2 - \left ( v, u \right )\left ( v, w \right )}{\left \| v \right \|^2 \left \| w \right \|^2 - \left ( v, w \right )^2}\]

Hence,

\begin{aligned} \left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left \| u \right \|^2 &= \left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left ( \left \| \alpha v + \beta w \right \|^2 + \left \| z \right \|^2 \right ) \\ &\geq \left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left ( \left \| \alpha v + \beta w \right \|^2 \right ) \\ &= \left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left ( \alpha^2 \left \| v \right \|^2 + \beta^2 \left \| w \right \|^2 + 2\alpha \beta \left ( v, w \right ) \right ) \\ &= \left ( w, u \right )^2 \left \| v \right \|^2 + \left ( v, u \right ) \left \| w \right \|^2 - 2 \left ( w, u \right ) \left ( v,u \right ) \left ( v, w \right ) \\ &= \left \| \left ( w, u \right ) v - \left ( v, u \right )w \right \|^2 \end{aligned}

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