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The Inner Product

The inner product (or ``dot product'', or ``scalar product'') is an operation on two vectors which produces a scalar. Defining an inner product for a Banach space specializes it to a Hilbert space (or ``inner product space''). There are many examples of Hilbert spaces, but we will only need $ \{{\bf C}^N,{\bf C}\}$ for this book (complex length $ N$ vectors, and complex scalars).

The inner product between (complex) $ N$-vectors $ \underline{u}$ and $ \underline{v}$ is defined by5.9

$\displaystyle \zbox {\left<\underline{u},\underline{v}\right> \isdef \sum_{n=0}^{N-1}u(n)\overline{v(n)}.}

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:5.10

$\displaystyle \left<\underline{u},\underline{u}\right> = \sum_{n=0}^{N-1}u(n)\o...
...sum_{n=0}^{N-1}\left\vert u(n)\right\vert^2 \isdef {\cal E}_u = \Vert u\Vert^2

As a result, the inner product is conjugate symmetric:

$\displaystyle \left<\underline{v},\underline{u}\right> = \overline{\left<\underline{u},\underline{v}\right>}

Note that the inner product takes $ {\bf C}^N\times{\bf C}^N$ to $ {\bf C}$. That is, two length $ N$ complex vectors are mapped to a complex scalar.

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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