Inner Product for Complex Vector Spaces

Definition

Let \(\mathbf{u}, \mathbf{v} \in \mathbb{C}^n\) be two complex vectors that

\[ \begin{align} \mathbf{u} &= \left( u_1, u_2, ..., u_n \right) \\ \mathbf{v} &= \left( v_1, v_2, ..., v_n \right) \end{align} \]

the inner product of \(\mathbf{u}\) and \(\mathbf{v}\) is defined as

\[ \left< \mathbf{u}, \mathbf{v} \right> = \sum_{i=1}^n u_i {v_i}^* \]

where \({v_i}^*\) is the conjugate of \(v_i\).

Properties

1. \(\left< \mathbf{v}, \mathbf{v} \right> \geq 0\)
2. \(\left< \mathbf{v}, \mathbf{v} \right> = 0 \Leftrightarrow \mathbf{v} = 0\)
3. \(\left< \mathbf{u}, \mathbf{v} + \mathbf{w} \right> = \left< \mathbf{u}, \mathbf{v} \right> + \left< \mathbf{u}, \mathbf{w} \right>\)
4. \(\left< \mathbf{u} + \mathbf{w}, \mathbf{v} \right> = \left< \mathbf{u}, \mathbf{v} \right> + \left< \mathbf{w}, \mathbf{v} \right>\)
5. For a scalar \(\alpha \in \mathbb{C}\), \(\left< \alpha \mathbf{u}, \mathbf{v} \right> = \alpha \left< \mathbf{u}, \mathbf{v} \right>\), \(\left< \mathbf{u}, \alpha \mathbf{v} \right> = \alpha^* \left< \mathbf{u}, \mathbf{v} \right>\)
6. \(\left< \mathbf{u}, \mathbf{v} \right> = {\left< \mathbf{v}, \mathbf{u} \right>}^*\)
7. \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal if \(\left< \mathbf{u}, \mathbf{v} \right> = 0\)
8. Inner product is nondegenerate
    # In other words, if \(\mathbf{x} \in \mathbb{C}^n\), and \(\left< \mathbf{x}, \mathbf{v} \right> = 0\) for all \(\mathbf{v} \in \mathbb{C}^n\), then \(\mathbf{x} = 0\)
9. Schwarz inequality: \(\left| \left< \mathbf{u}, \mathbf{v} \right> \right| \leq \left| \mathbf{u} \right| \left| \mathbf{v} \right|\)