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Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality (or ``Schwarz Inequality'') states that for all $ \underline{u}\in{\bf C}^N$ and $ \underline{v}\in{\bf C}^N$ , we have

$\displaystyle \zbox {\left\vert\left<\underline{u},\underline{v}\right>\right\vert \leq \Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert}
$

with equality if and only if $ \underline{u}=c\underline{v}$ for some scalar $ c$ .

We can quickly show this for real vectors $ \underline{u}\in{\bf R}^N$ , $ \underline{v}\in{\bf R}^N$ , as follows: If either $ \underline{u}$ or $ \underline{v}$ is zero, the inequality holds (as equality). Assuming both are nonzero, let's scale them to unit-length by defining the normalized vectors $ \underline{\tilde{u}}\isdeftext
\underline{u}/\Vert\underline{u}\Vert$ , $ \underline{\tilde{v}}\isdeftext \underline{v}/\Vert\underline{v}\Vert$ , which are unit-length vectors lying on the ``unit ball'' in $ {\bf R}^N$ (a hypersphere of radius $ 1$ ). We have

\begin{eqnarray*}
0 \leq \Vert\underline{\tilde{u}}-\underline{\tilde{v}}\Vert^2 &=& \left<\underline{\tilde{u}}-\underline{\tilde{v}},\underline{\tilde{u}}-\underline{\tilde{v}}\right> \\
&=& \left<\underline{\tilde{u}},\underline{\tilde{u}}-\underline{\tilde{v}}\right> - \left<\underline{\tilde{v}},\underline{\tilde{u}}-\underline{\tilde{v}}\right> \\
&=& \left<\underline{\tilde{u}},\underline{\tilde{u}}\right> - \left<\underline{\tilde{u}},\underline{\tilde{v}}\right> - \left<\underline{\tilde{v}},\underline{\tilde{u}}\right> + \left<\underline{\tilde{v}},\underline{\tilde{v}}\right> \\
&=& \Vert\underline{\tilde{u}}\Vert^2 - \left[\left<\underline{\tilde{u}},\underline{\tilde{v}}\right> + \overline{\left<\underline{\tilde{u}},\underline{\tilde{v}}\right>}\right]
+ \Vert\underline{\tilde{v}}\Vert^2 \\
&=& 2 - 2\mbox{re}\left\{\left<\underline{\tilde{u}},\underline{\tilde{v}}\right>\right\} \\
&=& 2 - 2\left<\underline{\tilde{u}},\underline{\tilde{v}}\right>
\end{eqnarray*}

which implies

$\displaystyle \left<\underline{\tilde{u}},\underline{\tilde{v}}\right> \leq 1
$

or, removing the normalization,

$\displaystyle \left<\underline{u},\underline{v}\right> \leq \Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert.
$

The same derivation holds if $ \underline{u}$ is replaced by $ -\underline{u}$ yielding

$\displaystyle -\left<\underline{u},\underline{v}\right> \leq \Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert.
$

The last two equations imply

$\displaystyle \left\vert\left<\underline{u},\underline{v}\right>\right\vert \leq \Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert.
$

In the complex case, let $ \left<\underline{u},\underline{v}\right>=R e^{j\theta}$ , and define $ \underline{\tilde{v}}=\underline{v}e^{j\theta}$ . Then $ \left<\underline{u},\underline{\tilde{v}}\right>$ is real and equal to $ \vert\left<\underline{u},\underline{\tilde{v}}\right>\vert=R>0$ . By the same derivation as above,

$\displaystyle \left<\underline{u},\underline{\tilde{v}}\right>\leq\Vert\underline{u}\Vert\cdot\Vert\underline{\tilde{v}}\Vert = \Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert.
$

Since $ \left<\underline{u},\underline{\tilde{v}}\right>=R=\left\vert\left<\underline{u},\underline{\tilde{v}}\right>\right\vert=\left\vert\left<\underline{u},\underline{v}\right>\right\vert$ , the result is established also in the complex case.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA