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

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 ). We have

$\begin{eqnarray*} 0 \leq \Vert\underline{\tilde{u}}-\underline{\tilde{v}}\Vert^2... ...=& 2 - 2\left<\underline{\tilde{u}},\underline{\tilde{v}}\right> \end{eqnarray*}$

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,

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

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Geometric Signal Theory
       The Inner Product
          Cauchy-Schwarz Inequality