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The Pythagorean Theorem in N-Space

In 2D, the Pythagorean Theorem says that when $ x$ and $ y$ are orthogonal, as in Fig.5.8, (i.e., when the vectors $ x$ and $ y$ intersect at a right angle), then we have

$\displaystyle \Vert x+y\Vert^2 = \Vert x\Vert^2 + \Vert y\Vert^2$   $\displaystyle \mbox{($x\perp y$)}$$\displaystyle . \protect$ (5.1)

This relationship generalizes to $ N$ dimensions, as we can easily show:
$\displaystyle \Vert x+y\Vert^2$ $\displaystyle =$ $\displaystyle \left<x+y,x+y\right>$  
  $\displaystyle =$ $\displaystyle \left<x,x\right>+\left<x,y\right>+\left<y,x\right>+\left<y,y\right>$  
  $\displaystyle =$ $\displaystyle \Vert x\Vert^2 + \left<x,y\right>+\overline{\left<x,y\right>} + \Vert y\Vert^2$  
  $\displaystyle =$ $\displaystyle \Vert x\Vert^2 + \Vert y\Vert^2 + 2$re$\displaystyle \left\{\left<x,y\right>\right\}
\protect$ (5.2)

If $ x\perp y$ , then $ \left<x,y\right>=0$ and Eq.$ \,$ (5.1) holds in $ N$ dimensions.

Note that the converse is not true in $ {\bf C}^N$ . That is, $ \Vert x+y\Vert^2 = \Vert x\Vert^2 + \Vert y\Vert^2$ does not imply $ x\perp y$ in $ {\bf C}^N$ . For a counterexample, consider $ x= (j,1)$ , $ y=
(1, -j)$ , in which case

$\displaystyle \Vert x+y\Vert^2 = \Vert 1+j,1-j\Vert^2 =
4 = \Vert x\Vert^2 + \Vert y\Vert^2
$

while $ \left<x,y\right> = j\cdot 1 + 1 \cdot\overline{-j} = 2j$ .

For real vectors $ x,y\in{\bf R}^N$ , the Pythagorean theorem Eq.$ \,$ (5.1) holds if and only if the vectors are orthogonal. To see this, note that, from Eq.$ \,$ (5.2), when the Pythagorean theorem holds, either $ x$ or $ y$ is zero, or $ \left<x,y\right>$ is zero or purely imaginary, by property 1 of norms (see §5.8.2). If the inner product cannot be imaginary, it must be zero.

Note that we also have an alternate version of the Pythagorean theorem:

$\displaystyle x\perp y\,\,\Rightarrow\,\,
\Vert x-y\Vert^2 = \Vert x\Vert^2 + \Vert y\Vert^2
$


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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
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