Cauchy-Schwarz Inequality

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

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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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