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Mathematics for Physics

242 Complex Conjugate

A mapping that maps a complex number to its conjugate value that is used to form real norms from complex scalars.

Complex conjugation of complex numbers produces real, non-negative values. This is useful for measuring lengths in complex vector spaces, such as Hilbert spaces. Complex conjugation can be applied to both complex numbers and complex vectors.

definition [d] (Complex Conjugate = Conjugate) A map \(\;\bar{\;\cdot\;}: \mathbb{C} \rightarrow \mathbb{C}\) that flips the imaginary part of a complex number:

  • \(\overline{a + bi} = a - bi\) .

where

  • \(a, b \in \mathbb{R}\).
  • \(i\) is the imaginary unit, \(i^2 = -1\).

Geometric interpretation of the complex conjugate: reflecting a complex number across the real axis in the complex plane.
Geometric interpretation of the complex conjugate: reflecting a complex number across the real axis in the complex plane.

242.1 Elementary Example

242.1.1 Simple

Complex conjugation flips the sign of the imaginary part.

\[ z = a + bi \]

\[ \overline{z} = a - bi \]

\[ z = 2 + 3i,\quad \overline{z} = 2 - 3i \]

where

  • \(a, b\) are real numbers.
  • \(i\) is the imaginary unit with \(i^{2} = -1\).
  • \(\overline{z}\) is the complex conjugate of \(z\).

242.1.2 General

Conjugation acts entrywise on a vector of complex numbers.

\[ v = (1+i,\ 2-3i,\ -i) \]

\[ \overline{v} = (1-i,\ 2+3i,\ i) \]

where

  • \(\overline{v}\) is the vector of conjugated components.

242.2 References

  1. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — definition.
  2. Cohen, M. X. Linear Algebra: Theory, Intuition, Code. Sincxpress BV, 2021. — definition; Figure 9.4, p. 245.