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)