Complex conjugate

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In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if and are real, then) the complex conjugate of is equal to The complex conjugate of is often denoted as

Geometric representation (Argand diagram) of and its conjugate in the complex plane. The complex conjugate is found by reflecting across the real axis.

In polar form, the conjugate of is This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number:  (or  in polar coordinates).

If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

NotationEdit

The complex conjugate of a complex number   is written as   or   The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics. If a complex number is represented as a   matrix, the notations are identical.[clarification needed]

PropertiesEdit

The following properties apply for all complex numbers   and   unless stated otherwise, and can be proved by writing   and   in the form  

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:[1]

 

A complex number is equal to its complex conjugate if its imaginary part is zero, or equivalently, if the number is real. In other words, real numbers are the only fixed points of conjugation.

Conjugation does not change the modulus of a complex number:  

Conjugation is an involution, that is, the conjugate of the conjugate of a complex number   is   In symbols,  [1]

The product of a complex number with its conjugate is equal to the square of the number's modulus. This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates.

 

Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:

 
 
 

If   is a polynomial with real coefficients and   then   as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).

In general, if   is a holomorphic function whose restriction to the real numbers is real-valued, and   and   are defined, then

 

The map   from   to   is a homeomorphism (where the topology on   is taken to be the standard topology) and antilinear, if one considers   as a complex vector space over itself. Even though it appears to be a well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension   This Galois group has only two elements:   and the identity on   Thus the only two field automorphisms of   that leave the real numbers fixed are the identity map and complex conjugation.

Use as a variableEdit

Once a complex number   or   is given, its conjugate is sufficient to reproduce the parts of the  -variable:

  • Real part:  
  • Imaginary part:  
  • Modulus (or absolute value):  
  • Argument:   so  

Furthermore,   can be used to specify lines in the plane: the set

 
is a line through the origin and perpendicular to   since the real part of   is zero only when the cosine of the angle between   and   is zero. Similarly, for a fixed complex unit   the equation
 
determines the line through   parallel to the line through 0 and  

These uses of the conjugate of   as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.

GeneralizationsEdit

The other planar real algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.

For matrices of complex numbers,   where   represents the element-by-element conjugation of  [2] Contrast this to the property   where   represents the conjugate transpose of  

Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.

One may also define a conjugation for quaternions and split-quaternions: the conjugate of   is  

All these generalizations are multiplicative only if the factors are reversed:

 

Since the multiplication of planar real algebras is commutative, this reversal is not needed there.

There is also an abstract notion of conjugation for vector spaces   over the complex numbers. In this context, any antilinear map   that satisfies

  1.   where   and   is the identity map on  
  2.   for all   and
  3.   for all  

is called a complex conjugation, or a real structure. As the involution   is antilinear, it cannot be the identity map on  

Of course,   is a  -linear transformation of   if one notes that every complex space   has a real form obtained by taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space  [3]

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no canonical notion of complex conjugation.

See alsoEdit

ReferencesEdit

  1. ^ a b Friedberg, Stephen; Insel, Arnold; Spence, Lawrence (2018), Linear Algebra (5 ed.), ISBN 978-0134860244, Appendix D
  2. ^ Arfken, Mathematical Methods for Physicists, 1985, pg. 201
  3. ^ Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29

BibliographyEdit

  • Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).