Geometric representation (
Argand diagram) of and its conjugate in the complex plane. The complex conjugate is found by
reflecting across the real axis.
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 numbers then the complex conjugate of is The complex conjugate of is often denoted as or .
In
polar form, if and are real numbers then 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).
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, and the complex conjugate corresponds to the
matrix transpose, which is a flip along the diagonal.[1]
Properties
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:[ref 1]
A complex number is equal to its complex conjugate if its imaginary part is zero, that is, 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, [ref 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:
[note 1]
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
fieldautomorphism. 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 variable
Once a complex number or is given, its conjugate is sufficient to reproduce the parts of the -variable:
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.
For matrices of complex numbers, where represents the element-by-element conjugation of [ref 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.
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 [2]
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 also
Absolute square – Product of a number by itselfPages displaying short descriptions of redirect targets