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:
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
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
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