The complex conjugate of a complex number is defined to be

where and are real numbers.

Geometrically, complex conjugation is reflection in the real axis of the complex plane. The points and have the same complex modulus and opposite imaginary parts.

The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).

The complex conjugate is implemented in the Wolfram Language as Conjugate[z].

Note that there are several notations in common use for the complex conjugate. Applied physics and engineering texts tend to prefer , while most modern math and theoretical physics texts favor . Unfortunately, the notation is also commonly used to denote adjoint operators and the conjugate transpose of a matrix. Because of these mutually contradictory conventions, care is needed when consulting the literature. In this work, is used to denote the complex conjugate.

Common notational conventions for complex conjugate are summarized in the table below.

In linear algebra, it is common to apply both the complex conjugate and transpose to the same matrix. The matrix obtained from a given matrix by this combined operation is commonly called the conjugate transpose of . It is also called the Hermitian conjugate or Hermitian adjoint and, in some contexts, the adjoint matrix. The adjugate matrix is a different operation: it is the transpose of the matrix of cofactors. In this work, denotes the conjugate transpose matrix, while denotes an adjoint operator when the context is operator theory.

By definition, the complex conjugate satisfies

The complex conjugate is distributive under complex addition,

since

and distributive over complex multiplication,

since

See also

Complex Addition, Complex Analysis, Complex Division, Complex Exponentiation, Complex Modulus, Complex Multiplication, Complex Number, Complex Subtraction, Conjugate Matrix, Conjugate Transpose Explore this topic in the MathWorld classroom

Related Wolfram sites

https://functions.wolfram.com/ComplexComponents/Conjugate/

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.Anton, H. Elementary Linear Algebra, 8th ed. New York: Wiley, 2000.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 355-356, 1985.Bekefi, G. and Barrett, A. H. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, p. 616, 1987.Golub, G. and Van Loan, C. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 21, 1998.Hecht, E. Optics, 3rd ed. Reading, MA: Addison-Wesley, p. 18, 1998.Herkommer, M. A. Number Theory: A Programmer's Guide. New York: McGraw-Hill, p. 262, 1999.Kaplan, W. Advanced Mathematics for Engineers. Reading, MA: Addison-Wesley, 1981.Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992.Krantz, S. G. "Complex Conjugate." §1.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 2, 1999.Kreyszig, E. Advanced Engineering Mathematics, 6th ed. New York: Wiley, p. 568, 1988.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, 1989.Roman, S. "The Conjugate of a Complex Number and Complex Division." §11.2 in College Algebra and Trigonometry. San Diego, CA: Harcourt, Brace, Jovanovich, pp. 534-541, 1987.Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993.

Referenced on Wolfram|Alpha

Complex Conjugate

Cite this as:

Weisstein, Eric W. "Complex Conjugate." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexConjugate.html

Subject classifications