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 $\mathbb{C}$ ,

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

A linear operator thus ``commutes with mixing.'' Linearity consists of two component properties:

• additivity: $f(\underline{u}+\underline{v}) = f(\underline{u}) + f(\underline{v})$
• homogeneity: $f(\alpha \underline{u}) = \alpha f(\underline{u})$

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 $\alpha, \beta\in\mathbb{C}$ ,

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

This is easy to show from the definition:

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

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

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

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

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

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

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

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.