Definition of conjugate of a vector in C^n

**Definition:** The *conjugate* (or *complex conjugate*) of a vector \(\mathbf{z} = (z_1, z_2, \ldots, z_n) \in \mathbb{C}^n\) is the vector obtained by taking the complex conjugate of each component:

\[\overline{\mathbf{z}} = (\overline{z_1}, \overline{z_2}, \ldots, \overline{z_n})\]

where \(\overline{a + bi} = a - bi\).

**Notation:** \(\overline{\mathbf{z}}\) or \(\mathbf{z}^*\) (the latter is common in physics and engineering).

**Examples:**
- \(\mathbf{z} = (1+i, 2-i, 3)\) → \(\overline{\mathbf{z}} = (1-i, 2+i, 3)\)
- \(\mathbf{w} = (i, -i, 0)\) → \(\overline{\mathbf{w}} = (-i, i, 0)\)

**Properties:**
- \(\overline{\overline{\mathbf{z}}} = \mathbf{z}\) (double conjugation returns the original)
- \(\overline{\mathbf{z}} = \mathbf{z}\) if and only if \(\mathbf{z} \in \mathbb{R}^n\)
- \(\overline{\mathbf{z} + \mathbf{w}} = \overline{\mathbf{z}} + \overline{\mathbf{w}}\) (conjugate of sum = sum of conjugates)Copy