Complex Inner Products
Table of Contents
1. Inner Product
An inner product satisfies the following basic properties:
- Conjugate commutivity: \(\langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle^*\)
- Scaling: \(\langle \alpha\mathbf{x}, \mathbf{y} \rangle = \alpha\langle\mathbf{x}, \mathbf{y}\rangle\)
- Nonnegativity: \(\langle \mathbf{x}, \mathbf{x} \rangle \geq 0\) with equality if and only if \(\mathbf{x} = 0\)
For real Cartesian spaces, we use the dot product, but this doesn't work for complex numbers.
2. Complex Inner Products
2.1. Inner Product for Vectors
If \(\mathbf{x}, \mathbf{y} \in \mathbb{C}^n\), then we define the complex inner product of these two vectors as the transpose of the first multiplied with the complex conjugate of the second:
\begin{align} \langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{x}^T\mathbf{y}^* = \begin{bmatrix} x_1 & \cdots & x_n \end{bmatrix}\begin{bmatrix}y_1^* \\ \vdots \\ y_n^*\end{bmatrix} = \sum_{k=1}^n x_ky_k^* \end{align}2.2. Inner Product for DT Signals
For signals \(x, y: \mathbb{Z} \rightarrow \mathbb{C}\):
\begin{align} \langle x, y \rangle = \sum_{n=-\infty}^{\infty}x[n]y^*[n] \end{align}2.3. Inner Product for CT Signals
For signals \(x, y: \mathbb{R} \rightarrow \mathbb{C}\):
\begin{align} \langle x, y \rangle = \int_{-\infty}^{\infty}x(t)y^*(t) \text{ d}t \end{align}