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

Created over 8 years ago, updated 3 days ago

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).