Cauchy-Schwarz Inequality

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Cauchy-Schwarz Inequality

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality (or “Schwarz Inequality”) states that for all $\ and $\, we have

\

with equality if and only if $\ for some scalar $c$.

We can quickly show this for real vectors $\, $\, as follows: If either $\ or $\ is zero, the inequality holds (as equality). Assuming both are nonzero, let’s scale them to unit-length by defining the normalized vectors ${\, ${\, which are unit-length vectors lying on the “unit ball” in ${\ (a hypersphere of radius $1$). We have

\


which implies
\

or, removing the normalization,
\

The same derivation holds if $\ is replaced by $-\ yielding
\

The last two equations imply
\

The complex case can be shown by rotating the components of $\ and $\such that $\ becomes equal to $\.

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