Definition of the imaginary part of a vector in C^n

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

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

where \(\text{Im}(a + bi) = b\) (the imaginary part is a **real** number, not \(bi\)).

**Alternative formula:** \(\text{Im}(\mathbf{z}) = \frac{\mathbf{z} - \overline{\mathbf{z}}}{2i}\)

**Examples:**
- \(\mathbf{z} = (1+i, 2-i, 3)\) → \(\text{Im}(\mathbf{z}) = (1, -1, 0)\)
- \(\mathbf{w} = (i, -2+3i, 5)\) → \(\text{Im}(\mathbf{w}) = (1, 3, 0)\)

**Decomposition:** Any \(\mathbf{z} \in \mathbb{C}^n\) can be written as:

\[\mathbf{z} = \text{Re}(\mathbf{z}) + i \cdot \text{Im}(\mathbf{z})\]

Note that \(\text{Re}(\mathbf{z}), \text{Im}(\mathbf{z}) \in \mathbb{R}^n\).Copy