Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

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

Created over 8 years ago, updated 3 days ago

Definition: The real 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 real part of each component:

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

where (\text{Re}(a + bi) = a).

Alternative formula: (\text{Re}(\mathbf{z}) = \frac{\mathbf{z} + \overline{\mathbf{z}}}{2})

Examples:

  • (\mathbf{z} = (1+i, 2-i, 3)) → (\text{Re}(\mathbf{z}) = (1, 2, 3))
  • (\mathbf{w} = (i, -2+3i, 5)) → (\text{Re}(\mathbf{w}) = (0, -2, 5))

Property: For any (\mathbf{z} \in \mathbb{C}^n):

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

This decomposes any complex vector into its real and imaginary parts.