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Linearity of the Inner Product

Any function $ f(\underline{u})$ of a vector $ \underline{u}\in\mathbb{C}^N$ (which we may call an operator on $ \mathbb{C}^N$ ) is said to be linear if for all $ \underline{u}\in\mathbb{C}^N$ and $ \underline{v}\in\mathbb{C}^N$ , and for all scalars $ \alpha$ and $ \beta $ in $ \mathbb{C}$ ,

$\displaystyle f(\alpha \underline{u}+ \beta \underline{v}) = \alpha f(\underline{u}) + \beta f(\underline{v}).

A linear operator thus ``commutes with mixing.'' Linearity consists of two component properties: A function of multiple vectors, e.g., $ f(\underline{u},\underline{v},\underline{w})$ can be linear or not with respect to each of its arguments.

The inner product $ \left<\underline{u},\underline{v}\right>$ is linear in its first argument, i.e., for all $ \underline{u},\underline{v},\underline{w}\in\mathbb{C}^N$ , and for all $ \alpha, \beta\in\mathbb{C}$ ,

$\displaystyle \left<\alpha \underline{u}+ \beta \underline{v},\underline{w}\right> = \alpha \left<\underline{u},\underline{w}\right> + \beta \left<\underline{v},\underline{w}\right>.

This is easy to show from the definition:

\left<\alpha \underline{u}+ \beta \underline{v},\underline{w}\right> &\isdef & \sum_{n=0}^{N-1}\left[\alpha u(n) + \beta v(n)\right]\overline{w(n)} \\
&=& \sum_{n=0}^{N-1}\alpha u(n)\overline{w(n)} + \sum_{n=0}^{N-1}\beta v(n)\overline{w(n)} \\
&=& \alpha \sum_{n=0}^{N-1}u(n)\overline{w(n)} + \beta \sum_{n=0}^{N-1}v(n)\overline{w(n)} \\
&\isdef & \alpha \left<\underline{u},\underline{w}\right> + \beta \left<\underline{v},\underline{w}\right>

The inner product is also additive in its second argument, i.e.,

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

but it is only conjugate homogeneous (or antilinear) in its second argument, since

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

The inner product is strictly linear in its second argument with respect to real scalars $ a$ and $ b$ :

$\displaystyle \left<\underline{u},a \underline{v}+ b \underline{w}\right> = a \left<\underline{u},\underline{v}\right> + b \left<\underline{u},\underline{w}\right>, \quad a,b\in\mathbb{R}

where $ \underline{u},\underline{v},\underline{w}\in\mathbb{C}^N$ .

Since the inner product is linear in both of its arguments for real scalars, it may be called a bilinear operator in that context.

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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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University